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id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #0 to #1: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='topology'>Topology</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/topology'>topology</a></strong> (<a class='existingWikiWord' href='/nlab/show/diff/general+topology'>point-set topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/point-free+topology'>point-free topology</a>)</p> <p>see also <em><a class='existingWikiWord' href='/nlab/show/diff/differential+topology'>differential topology</a></em>, <em><a class='existingWikiWord' href='/nlab/show/diff/algebraic+topology'>algebraic topology</a></em>, <em><a class='existingWikiWord' href='/nlab/show/diff/functional+analysis'>functional analysis</a></em> and <em><a class='existingWikiWord' href='/nlab/show/diff/topological+homotopy+theory'>topological</a> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a></em></p> <p><a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Topology'>Introduction</a></p> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subset</a>, <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subset</a>, <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>neighbourhood</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>, <a class='existingWikiWord' href='/nlab/show/diff/locale'>locale</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+base'>base for the topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/neighborhood+base'>neighbourhood base</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/finer+topology'>finer/coarser topology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closure</a>, <a class='existingWikiWord' href='/nlab/show/diff/interior'>interior</a>, <a class='existingWikiWord' href='/nlab/show/diff/boundary'>boundary</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/separation+axioms'>separation</a>, <a class='existingWikiWord' href='/nlab/show/diff/sober+topological+space'>sobriety</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a>, <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/uniformly+continuous+map'>uniformly continuous function</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/embedding+of+topological+spaces'>embedding</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/open+map'>open map</a>, <a class='existingWikiWord' href='/nlab/show/diff/closed+map'>closed map</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sequence'>sequence</a>, <a class='existingWikiWord' href='/nlab/show/diff/net'>net</a>, <a class='existingWikiWord' href='/nlab/show/diff/subnet'>sub-net</a>, <a class='existingWikiWord' href='/nlab/show/diff/filter'>filter</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/convergence'>convergence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/convenient+category+of+topological+spaces'>convenient category of topological spaces</a></li> </ul> </li> </ul> <p><strong><a href='Top#UniversalConstructions'>Universal constructions</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>initial topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>final topology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/subspace'>subspace</a>, <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient space</a>,</p> </li> <li> <p>fiber space, <a class='existingWikiWord' href='/nlab/show/diff/space+attachment'>space attachment</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product space</a>, <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union+topological+space'>disjoint union space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cylinder'>mapping cylinder</a>, <a class='existingWikiWord' href='/nlab/show/diff/cocylinder'>mapping cocylinder</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a>, <a class='existingWikiWord' href='/nlab/show/diff/mapping+cocone'>mapping cocone</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+telescope'>mapping telescope</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/colimits+of+normal+spaces'>colimits of normal spaces</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/stuff%2C+structure%2C+property'>Extra stuff, structure, properties</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/nice+topological+space'>nice topological space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/metric+space'>metric space</a>, <a class='existingWikiWord' href='/nlab/show/diff/metric+topology'>metric topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/metrisable+topological+space'>metrisable space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kolmogorov+topological+space'>Kolmogorov space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff space</a>, <a class='existingWikiWord' href='/nlab/show/diff/regular+space'>regular space</a>, <a class='existingWikiWord' href='/nlab/show/diff/normal+space'>normal space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sober+topological+space'>sober space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact space</a>, <a class='existingWikiWord' href='/nlab/show/diff/proper+map'>proper map</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+topological+space'>sequentially compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/countably+compact+topological+space'>countably compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+topological+space'>locally compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/sigma-compact+topological+space'>sigma-compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/paracompact+topological+space'>paracompact</a>, <a class='existingWikiWord' href='/nlab/show/diff/countably+paracompact+topological+space'>countably paracompact</a>, <a class='existingWikiWord' href='/nlab/show/diff/strongly+compact+topological+space'>strongly compact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compactly+generated+topological+space'>compactly generated space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/second-countable+space'>second-countable space</a>, <a class='existingWikiWord' href='/nlab/show/diff/first-countable+space'>first-countable space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/contractible+space'>contractible space</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+contractible+space'>locally contractible space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/connected+space'>connected space</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+connected+topological+space'>locally connected space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/simply+connected+space'>simply-connected space</a>, <a class='existingWikiWord' href='/nlab/show/diff/semi-locally+simply+connected+topological+space'>locally simply-connected space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cell+complex'>cell complex</a>, <a class='existingWikiWord' href='/nlab/show/diff/CW+complex'>CW-complex</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pointed+topological+space'>pointed space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+vector+space'>topological vector space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Banach+space'>Banach space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hilbert+space'>Hilbert space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+group'>topological group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+vector+bundle'>topological vector bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/topological+K-theory'>topological K-theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+manifold'>topological manifold</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/empty+space'>empty space</a>, <a class='existingWikiWord' href='/nlab/show/diff/point+space'>point space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/discrete+object'>discrete space</a>, <a class='existingWikiWord' href='/nlab/show/diff/codiscrete+space'>codiscrete space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Sierpinski+space'>Sierpinski space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/order+topology'>order topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/specialization+topology'>specialization topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/Scott+topology'>Scott topology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean space</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/real+number'>real line</a>, <a class='existingWikiWord' href='/nlab/show/diff/plane'>plane</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cylinder+object'>cylinder</a>, <a class='existingWikiWord' href='/nlab/show/diff/cone'>cone</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sphere'>sphere</a>, <a class='existingWikiWord' href='/nlab/show/diff/ball'>ball</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/circle'>circle</a>, <a class='existingWikiWord' href='/nlab/show/diff/torus'>torus</a>, <a class='existingWikiWord' href='/nlab/show/diff/annulus'>annulus</a>, <a class='existingWikiWord' href='/nlab/show/diff/M%C3%B6bius+strip'>Moebius strip</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/polytope'>polytope</a>, <a class='existingWikiWord' href='/nlab/show/diff/polyhedron'>polyhedron</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/projective+space'>projective space</a> (<a class='existingWikiWord' href='/nlab/show/diff/real+projective+space'>real</a>, <a class='existingWikiWord' href='/nlab/show/diff/complex+projective+space'>complex</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/classifying+space'>classifying space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/configuration+space+of+points'>configuration space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/path'>path</a>, <a class='existingWikiWord' href='/nlab/show/diff/loop'>loop</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>mapping spaces</a>: <a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>compact-open topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/topology+of+uniform+convergence'>topology of uniform convergence</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/loop+space'>loop space</a>, <a class='existingWikiWord' href='/nlab/show/diff/path+space'>path space</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Zariski+topology'>Zariski topology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Cantor+space'>Cantor space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Mandelbrot+set'>Mandelbrot space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Peano+curve'>Peano curve</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/line+with+two+origins'>line with two origins</a>, <a class='existingWikiWord' href='/nlab/show/diff/long+line'>long line</a>, <a class='existingWikiWord' href='/nlab/show/diff/Sorgenfrey+line'>Sorgenfrey line</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/K-topology'>K-topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/Dowker+space'>Dowker space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Warsaw+circle'>Warsaw circle</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hawaiian+earring+space'>Hawaiian earring space</a></p> </li> </ul> <p><strong>Basic statements</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+implies+sober'>Hausdorff spaces are sober</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/schemes+are+sober'>schemes are sober</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/continuous+images+of+compact+spaces+are+compact'>continuous images of compact spaces are compact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/closed+subspaces+of+compact+Hausdorff+spaces+are+equivalently+compact+subspaces'>closed subspaces of compact Hausdorff spaces are equivalently compact subspaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/open+subspaces+of+compact+Hausdorff+spaces+are+locally+compact'>open subspaces of compact Hausdorff spaces are locally compact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/quotient+projections+out+of+compact+Hausdorff+spaces+are+closed+precisely+if+the+codomain+is+Hausdorff'>quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+spaces+equivalently+have+converging+subnets'>compact spaces equivalently have converging subnet of every net</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lebesgue+number+lemma'>Lebesgue number lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+metric+spaces+are+equivalently+compact+metric+spaces'>sequentially compact metric spaces are equivalently compact metric spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+spaces+equivalently+have+converging+subnets'>compact spaces equivalently have converging subnet of every net</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+metric+spaces+are+totally+bounded'>sequentially compact metric spaces are totally bounded</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous'>continuous metric space valued function on compact metric space is uniformly continuous</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/paracompact+Hausdorff+spaces+are+normal'>paracompact Hausdorff spaces are normal</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/paracompact+Hausdorff+spaces+equivalently+admit+subordinate+partitions+of+unity'>paracompact Hausdorff spaces equivalently admit subordinate partitions of unity</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/closed+injections+are+embeddings'>closed injections are embeddings</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/proper+maps+to+locally+compact+spaces+are+closed'>proper maps to locally compact spaces are closed</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/injective+proper+maps+to+locally+compact+spaces+are+equivalently+the+closed+embeddings'>injective proper maps to locally compact spaces are equivalently the closed embeddings</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+compact+and+sigma-compact+spaces+are+paracompact'>locally compact and sigma-compact spaces are paracompact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+compact+and+second-countable+spaces+are+sigma-compact'>locally compact and second-countable spaces are sigma-compact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/second-countable+regular+spaces+are+paracompact'>second-countable regular spaces are paracompact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/CW-complexes+are+paracompact+Hausdorff+spaces'>CW-complexes are paracompact Hausdorff spaces</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Urysohn%27s+lemma'>Urysohn's lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Tietze+extension+theorem'>Tietze extension theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Tychonoff+theorem'>Tychonoff theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/tube+lemma'>tube lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Michael%27s+theorem'>Michael's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Brouwer%27s+fixed+point+theorem'>Brouwer's fixed point theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+invariance+of+dimension'>topological invariance of dimension</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Jordan+curve+theorem'>Jordan curve theorem</a></p> </li> </ul> <p><strong>Analysis Theorems</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Heine-Borel+theorem'>Heine-Borel theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/intermediate+value+theorem'>intermediate value theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/extreme+value+theorem'>extreme value theorem</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/topological+homotopy+theory'>topological homotopy theory</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy'>left homotopy</a>, <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>right homotopy</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+equivalence'>homotopy equivalence</a>, <a class='existingWikiWord' href='/nlab/show/diff/deformation+retract'>deformation retract</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+group'>fundamental group</a>, <a class='existingWikiWord' href='/nlab/show/diff/covering+space'>covering space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+theorem+of+covering+spaces'>fundamental theorem of covering spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+group'>homotopy group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/weak+homotopy+equivalence'>weak homotopy equivalence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Whitehead+theorem'>Whitehead's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Freudenthal+suspension+theorem'>Freudenthal suspension theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/nerve+theorem'>nerve theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+extension+property'>homotopy extension property</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+cofibration'>Hurewicz cofibration</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+cofiber+sequence'>cofiber sequence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Str%C3%B8m+model+structure'>Strøm model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+topological+spaces'>classical model structure on topological spaces</a></p> </li> </ul> </div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#statement'>Statement</a></li><li><a href='#related_concepts'>Related concepts</a></li></ul></div> <h2 id='statement'>Statement</h2> <div class='num_prop' id='OpenClosedContinuousInjectionsAreEmbeddings'> <h6 id='proposition'>Proposition</h6> <p><strong>(open/closed continuous injections are embeddings)</strong></p> <p>A <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a> <math class='maruku-mathml' display='inline' id='mathml_329fe4741f313410656cd22a5d84289dc9ceecda_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><msub><mi>τ</mi> <mi>X</mi></msub><mo stretchy='false'>)</mo><mo>→</mo><mo stretchy='false'>(</mo><mi>Y</mi><mo>,</mo><msub><mi>τ</mi> <mi>Y</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f \colon (X, \tau_X) \to (Y,\tau_Y)</annotation></semantics></math> which is</p> <ol> <li> <p>an <a class='existingWikiWord' href='/nlab/show/diff/injection'>injective function</a></p> </li> <li> <p>an <a class='existingWikiWord' href='/nlab/show/diff/open+map'>open map</a> or a <a class='existingWikiWord' href='/nlab/show/diff/closed+map'>closed map</a></p> </li> </ol> <p>is a <a class='existingWikiWord' href='/nlab/show/diff/embedding+of+topological+spaces'>topological embedding</a>.</p> </div> <div class='proof'> <h6 id='proof'>Proof</h6> <p>If <math class='maruku-mathml' display='inline' id='mathml_329fe4741f313410656cd22a5d84289dc9ceecda_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> is injective, then the map onto its <a class='existingWikiWord' href='/nlab/show/diff/image'>image</a> <math class='maruku-mathml' display='inline' id='mathml_329fe4741f313410656cd22a5d84289dc9ceecda_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>→</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>⊂</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>X \to f(X) \subset Y</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/bijection'>bijection</a>. Moreover, it is still continuous with respect to the <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>subspace topology</a> on <math class='maruku-mathml' display='inline' id='mathml_329fe4741f313410656cd22a5d84289dc9ceecda_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f(X)</annotation></semantics></math>. Now a bijective continuous function is a homeomorphism precisely if it is an <a class='existingWikiWord' href='/nlab/show/diff/open+map'>open map</a> or a <a class='existingWikiWord' href='/nlab/show/diff/closed+map'>closed map</a> (by <a href='homeomorphism#HomeoContinuousOpenBijection'>this prop.</a>). But the image projection of <math class='maruku-mathml' display='inline' id='mathml_329fe4741f313410656cd22a5d84289dc9ceecda_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> has this property, respectively, if <math class='maruku-mathml' display='inline' id='mathml_329fe4741f313410656cd22a5d84289dc9ceecda_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> does (by <a href='open+map#OpenClosedImageProjection'>this prop.</a>).</p> </div> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/proper+maps+to+locally+compact+spaces+are+closed'>proper maps to locally compact spaces are closed</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/maps+from+compact+spaces+to+Hausdorff+spaces+are+closed+and+proper'>maps from compact spaces to Hausdorff spaces are closed and proper</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/CW-complexes+are+paracompact+Hausdorff+spaces'>CW-complexes are Hausdorff spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+implies+sober'>Hausdorff spaces are sober</a>, <a class='existingWikiWord' href='/nlab/show/diff/schemes+are+sober'>schemes are sober</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/continuous+images+of+compact+spaces+are+compact'>continuous image of a compact space is compact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/closed+subspaces+of+compact+spaces+are+compact'>closed subsets of compact spaces are compact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+subspaces+of+Hausdorff+spaces+are+closed'>compact subspaces of Hausdorff spaces are closed</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/closed+subspaces+of+compact+Hausdorff+spaces+are+equivalently+compact+subspaces'>closed subspaces of compact Hausdorff spaces are equivalently compact subspaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/quotient+projections+out+of+compact+Hausdorff+spaces+are+closed+precisely+if+the+codomain+is+Hausdorff'>quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/paracompact+Hausdorff+spaces+are+normal'>paracompact Hausdorff spaces are normal</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/paracompact+Hausdorff+spaces+equivalently+admit+subordinate+partitions+of+unity'>paracompact Hausdorff spaces equivalently admit subordinate partitions of unity</a></p> </li> </ul> <p> </p> </div> <div class="revisedby"> <p> Created on May 12, 2017 at 22:45:25. 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