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class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle General Topology subsection</span> </button> <ul id="toc-General_Topology-sublist" class="vector-toc-list"> <li id="toc-Introduction_to_Topology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Introduction_to_Topology"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Introduction to Topology</span> </div> </a> <ul id="toc-Introduction_to_Topology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties_of_Topological_Spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Properties_of_Topological_Spaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Properties of Topological Spaces</span> </div> </a> <ul id="toc-Properties_of_Topological_Spaces-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Vector_Spaces" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vector_Spaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Vector Spaces</span> </div> </a> <ul id="toc-Vector_Spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algebraic_Topology" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Algebraic_Topology"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Algebraic Topology</span> </div> </a> <button aria-controls="toc-Algebraic_Topology-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Algebraic Topology subsection</span> </button> <ul id="toc-Algebraic_Topology-sublist" class="vector-toc-list"> <li id="toc-Homotopy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Homotopy"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Homotopy</span> </div> </a> <ul id="toc-Homotopy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polytopes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Polytopes"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Polytopes</span> </div> </a> <ul id="toc-Polytopes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Homology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Homology"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Homology</span> </div> </a> <ul id="toc-Homology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cohomology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cohomology"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Cohomology</span> </div> </a> <ul id="toc-Cohomology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Advanced_Methods" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Advanced_Methods"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Advanced Methods</span> </div> </a> <ul id="toc-Advanced_Methods-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Differential_Topology" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Differential_Topology"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Differential Topology</span> </div> </a> <ul id="toc-Differential_Topology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Appendices" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Appendices"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Appendices</span> </div> </a> <ul id="toc-Appendices-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Help" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Help"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Help</span> </div> </a> <button aria-controls="toc-Help-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Help subsection</span> </button> <ul id="toc-Help-sublist" class="vector-toc-list"> <li id="toc-Question_&_Answer" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Question_&_Answer"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Question & Answer</span> </div> </a> <ul id="toc-Question_&_Answer-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_Reading" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Further_Reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Further Reading</span> </div> </a> <ul id="toc-Further_Reading-sublist" class="vector-toc-list"> <li id="toc-General_Topology_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#General_Topology_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2.1</span> <span>General Topology</span> </div> </a> <ul id="toc-General_Topology_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algebraic_Topology_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Algebraic_Topology_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2.2</span> <span>Algebraic Topology</span> </div> </a> <ul id="toc-Algebraic_Topology_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> </ul> </div> </div> </nav> </div> 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href="https://de.wikibooks.org/wiki/Mathematik:_Topologie" title="Mathematik: Topologie – German" lang="de" hreflang="de" data-title="Mathematik: Topologie" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikibooks.org/wiki/Topologie" title="Topologie – French" lang="fr" hreflang="fr" data-title="Topologie" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikibooks.org/wiki/%E4%BD%8D%E7%9B%B8%E5%B9%BE%E4%BD%95%E5%AD%A6" title="位相幾何学 – Japanese" lang="ja" hreflang="ja" data-title="位相幾何学" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikibooks.org/wiki/Topologia_og%C3%B3lna" title="Topologia ogólna – Polish" lang="pl" hreflang="pl" data-title="Topologia ogólna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikibooks.org/wiki/Topologi" title="Topologi – Swedish" lang="sv" hreflang="sv" data-title="Topologi" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikibooks.org/wiki/Topoloji" title="Topoloji – Turkish" lang="tr" hreflang="tr" data-title="Topoloji" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a 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It is at its core a generalization of the concept of distance, though this will not be immediately apparent for the novice student. Topology generalizes many distance-related concepts, such as continuity, compactness, and convergence. </p> <table class="plainlinks noprint messagebox notice" style="width:250px; float:right; clear:right; margin:0px; margin-left:10px;"> <tbody><tr style="vertical-align:middle;"><td style="padding:0.1em; text-align:center; vertical-align:middle; width:45px; border:none;"> <span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Wikipedia-logo-v2.svg/40px-Wikipedia-logo-v2.svg.png" decoding="async" width="40" height="37" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Wikipedia-logo-v2.svg/60px-Wikipedia-logo-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/80/Wikipedia-logo-v2.svg/80px-Wikipedia-logo-v2.svg.png 2x" data-file-width="103" data-file-height="94" /></span></span></td> <td style="color:black; text-align:left; vertical-align:middle; padding:0.5em; padding-left:0em; border:none;"> <p><a href="https://en.wikipedia.org/wiki/" class="extiw" title="w:">Wikipedia</a> has related information at <a class="external text" href="https://en.wikipedia.org/wiki/Special:Search/Topology"><i><b>Topology</b></i></a> </p> </td></tr></tbody></table> <table class="plainlinks noprint messagebox growth" style="width:250px; 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margin-bottom: .5em; float: right; padding: .5em 0 .8em 1.4em; background: none; width: auto;"> <tbody><tr> <td><meta property="mw:PageProp/toc" /> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Before_You_Begin">Before You Begin</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topology&veaction=edit&section=1" title="Edit section: Before You Begin" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Topology&action=edit&section=1" title="Edit section's source code: Before You Begin"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In order to make things easier for you as a reader, as well as for the writers, you will be expected to be familiar with a few topics before beginning. </p> <ul><li><a href="/wiki/Real_analysis" class="mw-redirect" title="Real analysis">Real analysis</a> <ul><li><a href="https://en.wikipedia.org/wiki/Continuous_function" class="extiw" title="w:Continuous function">Continuous Functions</a></li> <li>Sequences & Series, <a href="https://en.wikipedia.org/wiki/Convergent_series" class="extiw" title="w:Convergent series">Convergence & Divergence</a></li></ul></li> <li><a href="/wiki/Topology/Set_Theory" class="mw-redirect" title="Topology/Set Theory">Set Theory</a> <ul><li>Set Operations: Union, Intersection, Complement, <a href="https://en.wikipedia.org/wiki/De_Morgan%27s_laws" class="extiw" title="w:De Morgan's laws">De Morgan's laws</a>, etc.</li> <li><a href="https://en.wikipedia.org/wiki/Order_theory" class="extiw" title="w:Order theory">Order Relations</a>: Ordered Sets, Equivalence relations, <a href="https://en.wikipedia.org/wiki/Lattice_(order)" class="extiw" title="w:Lattice (order)">Lattices</a>.</li> <li>Functions: Definition and Properties of Functions</li> <li><a href="https://en.wikipedia.org/wiki/Cardinality" class="extiw" title="w:Cardinality">Cardinality</a>: Finite, Countable, and Uncountable sets</li> <li><a href="https://en.wikipedia.org/wiki/Zorn%27s_Lemma" class="extiw" title="w:Zorn's Lemma">Zorn's Lemma</a> and the <a href="https://en.wikipedia.org/wiki/Axiom_of_Choice" class="extiw" title="w:Axiom of Choice">Axiom of Choice</a></li></ul></li> <li>Mathematical Logic & Proofs <ul><li>Mathematics is all about proofs. One of the goals of this book is to improve your skills in doing proofs, but you will not learn any of the basics here.</li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="Motivation_and_Preliminaries">Motivation and Preliminaries</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topology&veaction=edit&section=2" title="Edit section: Motivation and Preliminaries" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Topology&action=edit&section=2" title="Edit section's source code: Motivation and Preliminaries"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Chapter 1.1 <a href="/wiki/Topology/Introduction" title="Topology/Introduction">Introduction</a></li> <li>Chapter 1.2 <a href="/wiki/Topology/History" title="Topology/History">History</a></li> <li>Chapter 1.3 <a href="/wiki/Topology/Basic_Concepts_Set_Theory" title="Topology/Basic Concepts Set Theory">Basic Concepts Set Theory</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="50% developed  as of Apr 13, 2013"><img alt="50% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/9px-50_percents.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/14px-50_percents.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/18px-50_percents.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="General_Topology">General Topology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topology&veaction=edit&section=3" title="Edit section: General Topology" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Topology&action=edit&section=3" title="Edit section's source code: General Topology"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Introduction_to_Topology">Introduction to Topology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topology&veaction=edit&section=4" title="Edit section: Introduction to Topology" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Topology&action=edit&section=4" title="Edit section's source code: Introduction to Topology"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Chapter 2.2.1 <a href="/wiki/Topology/Metric_Spaces" title="Topology/Metric Spaces">Metric Spaces</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="75% developed  as of Apr 13, 2013"><img alt="75% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/75_percent.svg/9px-75_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/75_percent.svg/14px-75_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/75_percent.svg/18px-75_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 2.2.2 <a href="/wiki/Topology/Topological_Spaces" title="Topology/Topological Spaces">Topological Spaces</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="50% developed  as of Apr 13, 2013"><img alt="50% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/9px-50_percents.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/14px-50_percents.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/18px-50_percents.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 2.2.3 <a href="/wiki/Topology/Bases" title="Topology/Bases">Bases</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="50% developed  as of Apr 13, 2013"><img alt="50% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/9px-50_percents.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/14px-50_percents.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/18px-50_percents.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 2.2.4 <a href="/wiki/Topology/Points_in_Sets" title="Topology/Points in Sets">Points in Sets</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="50% developed  as of Apr 13, 2013"><img alt="50% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/9px-50_percents.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/14px-50_percents.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/18px-50_percents.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 2.2.5 <a href="/wiki/Topology/Sequences" title="Topology/Sequences">Sequences</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="25% developed  as of Apr 13, 2013"><img alt="25% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/9px-25_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/14px-25_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/18px-25_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 2.2.6 <a href="/wiki/Topology/Subspaces" title="Topology/Subspaces">Subspaces</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="25% developed  as of Apr 13, 2013"><img alt="25% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/9px-25_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/14px-25_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/18px-25_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 2.2.7 <a href="/wiki/Topology/Order" title="Topology/Order">Order</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="25% developed  as of Apr 13, 2013"><img alt="25% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/9px-25_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/14px-25_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/18px-25_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 2.2.8 <a href="/wiki/Topology/Order_Topology" title="Topology/Order Topology">Order Topology</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="25% developed  as of Apr 13, 2013"><img alt="25% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/9px-25_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/14px-25_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/18px-25_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 2.2.9 <a href="/wiki/Topology/Product_Spaces" title="Topology/Product Spaces">Product Spaces</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="50% developed  as of Apr 13, 2013"><img alt="50% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/9px-50_percents.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/14px-50_percents.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/18px-50_percents.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 2.2.10 <a href="/wiki/Topology/Quotient_Spaces" title="Topology/Quotient Spaces">Quotient Spaces</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="50% developed  as of Apr 13, 2013"><img alt="50% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/9px-50_percents.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/14px-50_percents.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/18px-50_percents.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 2.2.11 <a href="/wiki/Topology/Continuity_and_Homeomorphisms" title="Topology/Continuity and Homeomorphisms">Continuity and Homeomorphisms</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="25% developed  as of Apr 13, 2013"><img alt="25% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/9px-25_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/14px-25_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/18px-25_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Properties_of_Topological_Spaces">Properties of Topological Spaces</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topology&veaction=edit&section=5" title="Edit section: Properties of Topological Spaces" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Topology&action=edit&section=5" title="Edit section's source code: Properties of Topological Spaces"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Chapter 2.3.1 <a href="/wiki/Topology/Separation_Axioms" title="Topology/Separation Axioms">Separation Axioms</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="50% developed  as of Apr 13, 2013"><img alt="50% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/9px-50_percents.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/14px-50_percents.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/18px-50_percents.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 2.3.2 <a href="/wiki/Topology/Connectedness" title="Topology/Connectedness">Connectedness</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="50% developed  as of Apr 13, 2013"><img alt="50% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/9px-50_percents.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/14px-50_percents.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/18px-50_percents.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 2.3.3 <a href="/wiki/Topology/Path_Connectedness" title="Topology/Path Connectedness">Path Connectedness</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="50% developed  as of Apr 13, 2013"><img alt="50% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/9px-50_percents.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/14px-50_percents.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/18px-50_percents.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 2.3.4 <a href="/wiki/Topology/Compactness" title="Topology/Compactness">Compactness</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="50% developed  as of Apr 13, 2013"><img alt="50% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/9px-50_percents.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/14px-50_percents.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/18px-50_percents.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 2.3.5 <a href="/wiki/Topology/Comb_Space" title="Topology/Comb Space">Comb Space</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="50% developed  as of Apr 13, 2013"><img alt="50% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/9px-50_percents.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/14px-50_percents.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/18px-50_percents.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 2.3.6 <a href="/wiki/Topology/Local_Connectedness" title="Topology/Local Connectedness">Local Connectedness</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="50% developed  as of Apr 13, 2013"><img alt="50% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/9px-50_percents.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/14px-50_percents.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/18px-50_percents.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 2.3.7 <a href="/wiki/Topology/Linear_Continuum" title="Topology/Linear Continuum">Linear Continuum</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="50% developed  as of Apr 13, 2013"><img alt="50% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/9px-50_percents.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/14px-50_percents.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/18px-50_percents.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 2.3.8 <a href="/wiki/Topology/Countability" title="Topology/Countability">Countability</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="75% developed  as of Apr 13, 2013"><img alt="75% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/75_percent.svg/9px-75_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/75_percent.svg/14px-75_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/75_percent.svg/18px-75_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 2.3.9 <a href="/wiki/Topology/Cantor_Space" title="Topology/Cantor Space">Cantor Space</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="25% developed  as of Apr 13, 2013"><img alt="25% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/9px-25_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/14px-25_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/18px-25_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 2.3.10 <a href="/wiki/Topology/Completeness" title="Topology/Completeness">Completeness</a> - not a topological property <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="50% developed  as of Apr 13, 2013"><img alt="50% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/9px-50_percents.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/14px-50_percents.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/18px-50_percents.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 2.3.11 <a href="/wiki/Topology/Completion" title="Topology/Completion">Completion</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="25% developed  as of Apr 13, 2013"><img alt="25% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/9px-25_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/14px-25_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/18px-25_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 2.3.12 <a href="/wiki/Topology/Perfect_map" title="Topology/Perfect map">Perfect map</a> - optional section which is challenging <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="50% developed  as of Apr 13, 2013"><img alt="50% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/9px-50_percents.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/14px-50_percents.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/18px-50_percents.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Vector_Spaces">Vector Spaces</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topology&veaction=edit&section=6" title="Edit section: Vector Spaces" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Topology&action=edit&section=6" title="Edit section's source code: Vector Spaces"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Chapter 3.1 <a href="/wiki/Topology/Vector_Spaces" title="Topology/Vector Spaces">Vector Spaces</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="25% developed  as of Apr 13, 2013"><img alt="25% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/9px-25_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/14px-25_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/18px-25_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 3.2 <a href="/wiki/Topology/Morphisms" title="Topology/Morphisms">Morphisms</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="25% developed  as of Apr 13, 2013"><img alt="25% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/9px-25_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/14px-25_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/18px-25_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 3.3 <a href="/w/index.php?title=Topology/Convexity&action=edit&redlink=1" class="new" title="Topology/Convexity (does not exist)">Convexity</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed  as of Apr 13, 2013"><img alt="0% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 3.4 <a href="/w/index.php?title=Topology/Hahn-Banach_Theorem&action=edit&redlink=1" class="new" title="Topology/Hahn-Banach Theorem (does not exist)">Hahn-Banach Theorem</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed  as of Apr 13, 2013"><img alt="0% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 3.5 <a href="/w/index.php?title=Topology/Topological_Vector_Spaces&action=edit&redlink=1" class="new" title="Topology/Topological Vector Spaces (does not exist)">Topological Vector Spaces</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed  as of Apr 13, 2013"><img alt="0% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 3.6 <a href="/wiki/Topology/Euclidean_Spaces" title="Topology/Euclidean Spaces">Euclidean Spaces</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="50% developed  as of Apr 13, 2013"><img alt="50% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/9px-50_percents.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/14px-50_percents.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/18px-50_percents.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 3.7 <a href="/wiki/Topology/Normed_Vector_Spaces" title="Topology/Normed Vector Spaces">Normed Vector Spaces</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="25% developed  as of Apr 13, 2013"><img alt="25% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/9px-25_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/14px-25_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/18px-25_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 3.8 <a href="/wiki/Topology/Banach_Spaces" title="Topology/Banach Spaces">Banach Spaces</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="25% developed  as of Apr 13, 2013"><img alt="25% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/9px-25_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/14px-25_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/18px-25_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 3.9 <a href="/wiki/Topology/Hilbert_Spaces" title="Topology/Hilbert Spaces">Hilbert Spaces</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="25% developed  as of Apr 13, 2013"><img alt="25% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/9px-25_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/14px-25_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/18px-25_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Algebraic_Topology">Algebraic Topology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topology&veaction=edit&section=7" title="Edit section: Algebraic Topology" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Topology&action=edit&section=7" title="Edit section's source code: Algebraic Topology"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Homotopy">Homotopy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topology&veaction=edit&section=8" title="Edit section: Homotopy" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Topology&action=edit&section=8" title="Edit section's source code: Homotopy"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Chapter 4.1 <a href="/wiki/Topology/Free_group_and_presentation_of_a_group" title="Topology/Free group and presentation of a group">Free group and presentation of a group</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="75% developed  as of Apr 13, 2013"><img alt="75% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/75_percent.svg/9px-75_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/75_percent.svg/14px-75_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/75_percent.svg/18px-75_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 4.2 <a href="/wiki/Topology/Deformation_Retract" title="Topology/Deformation Retract">Deformation Retract</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="25% developed  as of Apr 13, 2013"><img alt="25% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/9px-25_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/14px-25_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/18px-25_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 4.3 <a href="/wiki/Topology/Homotopy" title="Topology/Homotopy">Homotopy</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="50% developed  as of Apr 13, 2013"><img alt="50% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/9px-50_percents.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/14px-50_percents.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/18px-50_percents.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 4.4 <a href="/wiki/Topology/The_fundamental_group" title="Topology/The fundamental group">The fundamental group</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="75% developed  as of Apr 13, 2013"><img alt="75% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/75_percent.svg/9px-75_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/75_percent.svg/14px-75_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/75_percent.svg/18px-75_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 4.5 <a href="/wiki/Topology/Induced_homomorphism" title="Topology/Induced homomorphism">Induced homomorphism</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="50% developed  as of Apr 13, 2013"><img alt="50% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/9px-50_percents.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/14px-50_percents.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/18px-50_percents.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Polytopes">Polytopes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topology&veaction=edit&section=9" title="Edit section: Polytopes" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Topology&action=edit&section=9" title="Edit section's source code: Polytopes"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Chapter 5.1 <a href="/wiki/Topology/Simplicial_complexes" title="Topology/Simplicial complexes">Simplicial complexes</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="25% developed  as of Apr 13, 2013"><img alt="25% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/9px-25_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/14px-25_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/18px-25_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 5.2 <a href="/w/index.php?title=Topology/Barycentric_Coordinates&action=edit&redlink=1" class="new" title="Topology/Barycentric Coordinates (does not exist)">Barycentric Coordinates</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed  as of Apr 13, 2013"><img alt="0% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 5.3 <a href="/w/index.php?title=Topology/Geometric_Complexes&action=edit&redlink=1" class="new" title="Topology/Geometric Complexes (does not exist)">Geometric Complexes</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed  as of Apr 13, 2013"><img alt="0% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 5.4 <a href="/w/index.php?title=Topology/Barycentric_Subdivision&action=edit&redlink=1" class="new" title="Topology/Barycentric Subdivision (does not exist)">Barycentric Subdivision</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed  as of Apr 13, 2013"><img alt="0% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 5.5 <a href="/w/index.php?title=Topology/Simplical_Mappings&action=edit&redlink=1" class="new" title="Topology/Simplical Mappings (does not exist)">Simplical Mappings</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed  as of Apr 13, 2013"><img alt="0% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 5.6 <a href="/w/index.php?title=Topology/Imbedding_Theorem&action=edit&redlink=1" class="new" title="Topology/Imbedding Theorem (does not exist)">Imbedding Theorem</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed  as of Apr 13, 2013"><img alt="0% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Homology">Homology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topology&veaction=edit&section=10" title="Edit section: Homology" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Topology&action=edit&section=10" title="Edit section's source code: Homology"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Chapter 6.1 <a href="/wiki/Topology/Exact_Sequences" title="Topology/Exact Sequences">Exact Sequences</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="25% developed  as of Apr 13, 2013"><img alt="25% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/9px-25_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/14px-25_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/18px-25_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 6.2 <a href="/wiki/Topology/Homology_Groups" title="Topology/Homology Groups">Homology Groups</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="25% developed  as of Apr 13, 2013"><img alt="25% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/9px-25_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/14px-25_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/18px-25_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 6.3 <a href="/wiki/Topology/Singular_Homology" title="Topology/Singular Homology">Singular Homology</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed  as of Apr 13, 2013"><img alt="0% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 6.4 <a href="/wiki/Topology/Relative_Homology" title="Topology/Relative Homology">Relative Homology</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="25% developed  as of Apr 13, 2013"><img alt="25% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/9px-25_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/14px-25_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/18px-25_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 6.5 <a href="/wiki/Topology/Mayer-Vietoris_Sequence" title="Topology/Mayer-Vietoris Sequence">Mayer-Vietoris Sequence</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="25% developed  as of Apr 13, 2013"><img alt="25% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/9px-25_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/14px-25_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/18px-25_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 6.6 <a href="/w/index.php?title=Topology/Excision_Theorem&action=edit&redlink=1" class="new" title="Topology/Excision Theorem (does not exist)">Excision Theorem</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed  as of Apr 13, 2013"><img alt="0% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 6.7 <a href="/wiki/Topology/Eilenberg%E2%80%93Steenrod_axioms" title="Topology/Eilenberg–Steenrod axioms">Eilenberg–Steenrod axioms</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed  as of Apr 13, 2013"><img alt="0% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 6.8 <a href="/w/index.php?title=Topology/Relative_Homotopy&action=edit&redlink=1" class="new" title="Topology/Relative Homotopy (does not exist)">Relative Homotopy</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed  as of Apr 13, 2013"><img alt="0% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 6.9 <a href="/w/index.php?title=Topology/Vietoris_Homology&action=edit&redlink=1" class="new" title="Topology/Vietoris Homology (does not exist)">Vietoris Homology</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed  as of Apr 13, 2013"><img alt="0% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Cohomology">Cohomology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topology&veaction=edit&section=11" title="Edit section: Cohomology" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Topology&action=edit&section=11" title="Edit section's source code: Cohomology"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Chapter 7.1 <a href="/wiki/Topology/Cohomology" title="Topology/Cohomology">Cohomology</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed  as of Apr 13, 2013"><img alt="0% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 7.2 <a href="/w/index.php?title=Topology/Cohomology_Product&action=edit&redlink=1" class="new" title="Topology/Cohomology Product (does not exist)">Cohomology Product</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed  as of Apr 13, 2013"><img alt="0% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 7.3 <a href="/w/index.php?title=Topology/Cap-Product&action=edit&redlink=1" class="new" title="Topology/Cap-Product (does not exist)">Cap-Product</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed  as of Apr 13, 2013"><img alt="0% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 7.4 <a href="/w/index.php?title=Topology/Relative_Cohomology&action=edit&redlink=1" class="new" title="Topology/Relative Cohomology (does not exist)">Relative Cohomology</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed  as of Apr 13, 2013"><img alt="0% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 7.5 <a href="/w/index.php?title=Topology/Induced_Homeomorphism&action=edit&redlink=1" class="new" title="Topology/Induced Homeomorphism (does not exist)">Induced Homeomorphism</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed  as of Apr 13, 2013"><img alt="0% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 7.6 <a href="/w/index.php?title=Topology/%C4%8Cech_Cohomology&action=edit&redlink=1" class="new" title="Topology/Čech Cohomology (does not exist)">Čech Cohomology</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed  as of Apr 13, 2013"><img alt="0% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Advanced_Methods">Advanced Methods</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topology&veaction=edit&section=12" title="Edit section: Advanced Methods" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Topology&action=edit&section=12" title="Edit section's source code: Advanced Methods"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Chapter 8.1 <a href="/w/index.php?title=Topology/Poincar%C3%A9_Duality&action=edit&redlink=1" class="new" title="Topology/Poincaré Duality (does not exist)">Poincaré Duality</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed  as of Apr 13, 2013"><img alt="0% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 8.2 <a href="/w/index.php?title=Topology/Spectral_Sequences&action=edit&redlink=1" class="new" title="Topology/Spectral Sequences (does not exist)">Spectral Sequences</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed  as of Apr 13, 2013"><img alt="0% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Differential_Topology">Differential Topology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topology&veaction=edit&section=13" title="Edit section: Differential Topology" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Topology&action=edit&section=13" title="Edit section's source code: Differential Topology"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Chapter 9.1 <a href="/wiki/Topology/Manifolds" title="Topology/Manifolds">Manifolds</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="50% developed  as of Apr 13, 2013"><img alt="50% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/9px-50_percents.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/14px-50_percents.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/18px-50_percents.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 9.1.1 <a href="/wiki/Topology/Manifolds/Categories_of_Manifolds" title="Topology/Manifolds/Categories of Manifolds">Categories of Manifolds</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="50% developed  as of Apr 13, 2013"><img alt="50% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/9px-50_percents.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/14px-50_percents.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/18px-50_percents.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 9.2 <a href="/wiki/Topology/Tangent_Spaces" title="Topology/Tangent Spaces">Tangent Spaces</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="50% developed  as of Apr 13, 2013"><img alt="50% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/9px-50_percents.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/14px-50_percents.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/18px-50_percents.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li> <li>Chapter 9.3 <a href="/wiki/Topology/Vector_Bundles" title="Topology/Vector Bundles">Vector Bundles</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="25% developed  as of Apr 13, 2013"><img alt="25% developed  as of Apr 13, 2013" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/9px-25_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/14px-25_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/18px-25_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span><small> (Apr 13, 2013)</small></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Appendices">Appendices</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topology&veaction=edit&section=14" title="Edit section: Appendices" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Topology&action=edit&section=14" title="Edit section's source code: Appendices"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>A <a href="/w/index.php?title=Topology/Further_Reading&action=edit&redlink=1" class="new" title="Topology/Further Reading (does not exist)">Further Reading</a></li> <li>B <a href="/wiki/Topology/Index" title="Topology/Index">Index</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Help">Help</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topology&veaction=edit&section=15" title="Edit section: Help" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Topology&action=edit&section=15" title="Edit section's source code: Help"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Question_&_Answer"><span id="Question_.26_Answer"></span>Question & Answer</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topology&veaction=edit&section=16" title="Edit section: Question & Answer" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Topology&action=edit&section=16" title="Edit section's source code: Question & Answer"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Have a question? Why not ask the very textbook that you are learning from? </p><p>1. What is the difference between topology, algebra and analysis? </p> <ul><li>Topology is a generalization of analysis and geometry. It comes in many flavors: point-set topology, manifold topology and algebraic topology, to name a few. All topology generalizes concepts from analysis dealing with space such as continuity of functions, connectedness of a space, open and closed sets, (etc.). Algebraic topology attributes algebraic structures (groups, rings etc.) to families of topological spaces to distinguish topological differences in those families. Manifold topology works with spaces that are locally the same as Euclidean space, i.e., surfaces. Often manifolds are equipped with extra structure, such as smooth, PL, symplectic, etc. A naive description of topology is that it identifies those qualities of a space that do not change under twisting and stretching of that space. As such, it is popularly referred to as "rubber sheet geometry." In reality topology does far more than this, in fact providing a rigorous foundation under all branches of mathematics dealing with "spaces."</li></ul> <ul><li>Algebra deals with the structure of sets under various operations with particular properties. Commonly studied algebraic objects include Groups, Rings and Field. One of the major results from Algebra include Galois Theory, which eventually shows that there is no general solution to quintic polynomial equations by radicals. Also important results from Algebra are the Fundamental Theorem of Algebra (which says that, in the Field of Complex numbers, every non constant polynomial has at least one root), Group Classification, and much more.</li></ul> <ul><li>Analysis (or specifically real analysis) on the other hand deals with the real numbers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> and the standard topology and algebraic structure of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>. Analysis provides rigorous proofs for the definitions of derivatives and integrals, as well as treatments of sequences and limits. One can, in some sense, view it as a rigorous treatment of the Calculus.</li></ul> <p><br /> 2. How are the concepts of base and open cover related? It seems that every base is an open cover, but not every open cover is a base. But, why are both concepts needed? </p> <ul><li>The terms base and open cover are not evidently related. Every base is an open cover which is probably the main relation. Take a second countable topological space for instance (second countable means that the space has a countable base for its topology). Such a space satisfies the property that every open cover has a countable sub-cover. To prove this we use the countability of the base. Basically, for any open cover, we choose for each element of the space, an element of the open cover containing it and hence a basis element contained in that element of open cover. Therefore, for any open cover, we can generate a open cover of basis elements that is an 'open refinement' (see Wikipedia for definition). From here we can get properties of open covers from properties of the base. If the base is countable, we can generate a countable open cover from the original cover.</li></ul> <p>The reason we have both definitions is because these two things have different properties. The most useful fact about a base is that it determines the topology. A basis must have "arbitrarily small" sets, that is, any open set contains a basis element. On the other hand, an open cover does not determine the topology at all. It can be used to build things such as partitions of unity, and often draws on the compactness property. <a href="/wiki/User:Topology_Expert" title="User:Topology Expert">Topology Expert</a> (<a href="/wiki/User_talk:Topology_Expert" title="User talk:Topology Expert">talk</a>) 04:17, 8 June 2008 (UTC) </p><p>3. What is a homology? </p> <ul><li>While homotopy is what one would want to study at first, it turns out that most questions in homotopy theory are rather hard. If we "replace" the spheres of homotopy theory with simplices we can extract similar information about "holes" in the space (often what one is interested in), we get a much more computable sequence of groups. The Hurewicz theorem even gives us that in some cases the homotopy groups can be calculated via the homology groups.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Further_Reading">Further Reading</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topology&veaction=edit&section=17" title="Edit section: Further Reading" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Topology&action=edit&section=17" title="Edit section's source code: Further Reading"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="plainlinks noprint messagebox notice" style="width:250px; float:right; clear:right; margin:0px; margin-left:10px;"> <tbody><tr style="vertical-align:middle;"><td style="padding:0.1em; text-align:center; vertical-align:middle; width:45px; border:none;"> <span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></td> <td style="color:black; text-align:left; vertical-align:middle; padding:0.5em; padding-left:0em; border:none;"> <p><a href="https://commons.wikimedia.org/wiki/" class="extiw" title="commons:">Wikimedia Commons</a> has media related to: <i><b><a class="external text" href="https://commons.wikimedia.org/wiki/Special:Search/Topology">Topology</a></b></i> </p> </td></tr></tbody></table> <table class="plainlinks noprint messagebox notice" style="width:250px; float:right; clear:right; margin:0px; margin-left:10px;"> <tbody><tr style="vertical-align:middle;"><td style="padding:0.1em; text-align:center; vertical-align:middle; width:45px; border:none;"> <span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Wikipedia-logo-v2.svg/40px-Wikipedia-logo-v2.svg.png" decoding="async" width="40" height="37" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Wikipedia-logo-v2.svg/60px-Wikipedia-logo-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/80/Wikipedia-logo-v2.svg/80px-Wikipedia-logo-v2.svg.png 2x" data-file-width="103" data-file-height="94" /></span></span></td> <td style="color:black; text-align:left; vertical-align:middle; padding:0.5em; padding-left:0em; border:none;"> <p><a href="https://en.wikipedia.org/wiki/" class="extiw" title="w:">Wikipedia</a> has related information at <a class="external text" href="https://en.wikipedia.org/wiki/Special:Search/Topology"><i><b>Topology</b></i></a> </p> </td></tr></tbody></table> <div class="mw-heading mw-heading4"><h4 id="General_Topology_2">General Topology</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topology&veaction=edit&section=18" title="Edit section: General Topology" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Topology&action=edit&section=18" title="Edit section's source code: General Topology"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Aleksandrov; <i>Combinatorial Topology</i> (1956)</li> <li>Baker; <i>Introduction to Topology</i> (1991)</li> <li>Dixmier; <i>General Topology</i> (1984)</li> <li>Engelking; <i>General Topology</i> (1977)</li> <li>Munkres; <i>Topology</i> (2000)</li> <li>James; <i>Topological and Uniform Spaces</i> (1987)</li> <li>Jänich; <i>Topology</i> (1984)</li> <li>Kuratowski; <i>Introduction to Set Theory and Topology</i> (1961)</li> <li>Kuratowski; <i>Topology</i> (1966)</li> <li>Roseman; <i>Elementary Topology</i> (1999)</li> <li>Seebach, Steen; <i><a href="https://en.wikipedia.org/wiki/Counterexamples_in_Topology" class="extiw" title="w:Counterexamples in Topology">Counterexamples in Topology</a></i> (1978)</li> <li>Willard; <i>General Topology</i> (1970)</li></ul> <div class="mw-heading mw-heading4"><h4 id="Algebraic_Topology_2">Algebraic Topology</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topology&veaction=edit&section=19" title="Edit section: Algebraic Topology" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Topology&action=edit&section=19" title="Edit section's source code: Algebraic Topology"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Marvin Greenberg and John Harper; <i>Algebraic Topology</i> (1981)</li> <li>Allen Hatcher, <i>Algebraic Topology</i> (2002) <a rel="nofollow" class="external autonumber" href="http://www.math.cornell.edu/~hatcher/#ATI">[1]</a></li> <li>Hu, Sze-tsen, <i>Cohomology Theory</i> (1968)</li> <li>Hu, Sze-tsen, <i>Homology Theory</i> (1966)</li> <li>Hu, Sze-tsen, <i>Homotopy Theory</i> (1959)</li> <li>Albert T. Lundell and Stephen Weingram, <i>The Topology of CW Complexes</i> (1969)</li> <li>Joerg Mayer, <i>Algebraic Topology</i> (1972)</li> <li>James Munkres, <i>Elements of Algebraic Topology</i> (1984)</li> <li>Joseph J. Rotman, <i>An Introduction to Algebraic Topology</i> (1988)</li> <li>Edwin Spanier, <i>Algebraic Topology</i> (1966)</li></ul></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikibooks.org/w/index.php?title=Topology&oldid=4018877">https://en.wikibooks.org/w/index.php?title=Topology&oldid=4018877</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Special:Categories" title="Special:Categories">Categories</a>: <ul><li><a href="/wiki/Category:Book:Topology" title="Category:Book:Topology">Book:Topology</a></li><li><a href="/wiki/Category:Shelf:Topology" title="Category:Shelf:Topology">Shelf:Topology</a></li><li><a href="/wiki/Category:Shelf:University_level_mathematics_books" title="Category:Shelf:University level mathematics books">Shelf:University level mathematics books</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Books_with_Public_Collections" title="Category:Books with Public Collections">Books with Public Collections</a></li><li><a href="/wiki/Category:Subject:Topology" title="Category:Subject:Topology">Subject:Topology</a></li><li><a href="/wiki/Category:Books_with_book_name_as_subject" title="Category:Books with book name as subject">Books with book name as subject</a></li><li><a href="/wiki/Category:Subject:Topology/all_books" title="Category:Subject:Topology/all books">Subject:Topology/all books</a></li><li><a href="/wiki/Category:Subject:Pure_mathematics/all_books" title="Category:Subject:Pure mathematics/all books">Subject:Pure mathematics/all books</a></li><li><a href="/wiki/Category:Subject:Mathematics/all_books" title="Category:Subject:Mathematics/all books">Subject:Mathematics/all books</a></li><li><a href="/wiki/Category:Subject:Books_by_subject/all_books" title="Category:Subject:Books by subject/all books">Subject:Books by subject/all books</a></li><li><a href="/wiki/Category:Subject:University_level_mathematics_books" title="Category:Subject:University level mathematics books">Subject:University level mathematics books</a></li><li><a href="/wiki/Category:Subject:University_level_mathematics_books/all_books" title="Category:Subject:University level mathematics books/all books">Subject:University level mathematics books/all books</a></li><li><a href="/wiki/Category:Book:Wikibooks_Stacks/Books" title="Category:Book:Wikibooks Stacks/Books">Book:Wikibooks Stacks/Books</a></li><li><a href="/wiki/Category:Shelf:Topology/all_books" title="Category:Shelf:Topology/all books">Shelf:Topology/all books</a></li><li><a href="/wiki/Category:Department:Mathematics/all_books" title="Category:Department:Mathematics/all books">Department:Mathematics/all books</a></li><li><a href="/wiki/Category:Shelf:Pure_mathematics/all_books" title="Category:Shelf:Pure mathematics/all books">Shelf:Pure mathematics/all books</a></li><li><a href="/wiki/Category:Shelf:University_level_mathematics_books/all_books" title="Category:Shelf:University level mathematics books/all books">Shelf:University level mathematics books/all books</a></li><li><a href="/wiki/Category:Alphabetical/T" title="Category:Alphabetical/T">Alphabetical/T</a></li><li><a href="/wiki/Category:Half-finished_books" title="Category:Half-finished books">Half-finished books</a></li><li><a href="/wiki/Category:Books_by_completion_status/all_books" title="Category:Books by completion status/all books">Books by completion status/all books</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 24 December 2021, at 13:03.</li> <li id="footer-info-copyright">Text is available under the <a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike License</a>; additional terms may apply. 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