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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 #2 to #3: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a><ul><li><a href='#in_analysis_and_cohomology'><span> In<del class='diffmod'> analysis</del><ins class='diffmod'> set</ins><del class='diffmod'> and</del><ins class='diffmod'> theory</ins><del class='diffdel'> cohomology</del></span></a></li><ins class='diffins'><li><a href='#in_topology'>In topology</a></li></ins><li><a href='#in_topos_theory_and_type_theory'>In topos theory and type theory</a></li></ul></li><li><a href='#related_concepts'>Related concepts</a></li><ins class='diffins'><li><a href='#references'>References</a></li></ins></ul></div> <h2 id='idea'>Idea</h2> <p>The term “support” means different things in different parts of mathematics:</p> <h3 id='in_analysis_and_cohomology'><span> In<del class='diffmod'> analysis</del><ins class='diffmod'> set</ins><del class='diffmod'> and</del><ins class='diffmod'> theory</ins><del class='diffdel'> cohomology</del></span></h3> <p>Given <math class='maruku-mathml' display='inline' id='mathml_ee75fd3d4669d65998684d3185266a5b020a90c1_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><del class='diffmod'><mo>:</mo></del><ins class='diffmod'><mo lspace='verythinmathspace'>:</mo></ins><mi>X</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'><span> f<del class='diffmod'> :</del><ins class='diffmod'> \colon</ins> X \to A</span></annotation></semantics></math><span> <del class='diffmod'> some</del><ins class='diffmod'> a</ins><del class='diffdel'> quantity</del><del class='diffdel'> (a</del></span><a class='existingWikiWord' href='/nlab/show/diff/function'>function</a><span> <del class='diffmod'> or</del><ins class='diffmod'> to</ins><del class='diffdel'> more</del><del class='diffdel'> generally</del><del class='diffdel'> some</del></span><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/cocycle'>cocycle</a></del><ins class='diffmod'><math class='maruku-mathml' display='inline' id='mathml_b22a0abc0eaf50c4553d1069aa296b6ead2010ed_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math></ins><span><del class='diffmod'> )</del><ins class='diffmod'> </ins><del class='diffmod'> and</del><ins class='diffmod'> a</ins><del class='diffdel'> for</del></span><del class='diffdel'><math class='maruku-mathml' display='inline' id='mathml_ee75fd3d4669d65998684d3185266a5b020a90c1_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math></del><del class='diffdel'> a </del><a class='existingWikiWord' href='/nlab/show/diff/pointed+object'>pointed object</a> with the point playing the role of a <a class='existingWikiWord' href='/nlab/show/diff/zero'>zero</a> <a class='existingWikiWord' href='/nlab/show/diff/generalized+element'>element</a><span> ,<ins class='diffins'> then</ins> the</span><em>support</em> of <math class='maruku-mathml' display='inline' id='mathml_ee75fd3d4669d65998684d3185266a5b020a90c1_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/subobject'>subobject</a> of <math class='maruku-mathml' display='inline' id='mathml_ee75fd3d4669d65998684d3185266a5b020a90c1_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> on which <math class='maruku-mathml' display='inline' id='mathml_ee75fd3d4669d65998684d3185266a5b020a90c1_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> is non-zero.</p> <ins class='diffins'><h3 id='in_topology'>In topology</h3></ins><ins class='diffins'> </ins><ins class='diffins'><p>In <a class='existingWikiWord' href='/nlab/show/diff/topology'>topology</a> the support of a <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a> <math class='maruku-mathml' display='inline' id='mathml_b22a0abc0eaf50c4553d1069aa296b6ead2010ed_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>f \colon X \to A</annotation></semantics></math> as above is usually taken to be the <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>topological closure</a> of the set of point on which <math class='maruku-mathml' display='inline' id='mathml_b22a0abc0eaf50c4553d1069aa296b6ead2010ed_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> vanishes</p></ins><ins class='diffins'> </ins><ins class='diffins'><div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_b22a0abc0eaf50c4553d1069aa296b6ead2010ed_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Supp</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>Cl</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mi>x</mi><mo>∈</mo><mi>X</mi><mo stretchy='false'>|</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>≠</mo><mn>0</mn><mo>∈</mo><mi>A</mi><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> Supp(f) = Cl(\{x \in X \vert f(x) \neq 0 \in A\}) \,. </annotation></semantics></math></div></ins><ins class='diffins'> </ins><h3 id='in_topos_theory_and_type_theory'>In topos theory and type theory</h3> <p>support = <a class='existingWikiWord' href='/nlab/show/diff/n-truncated+object+of+an+%28infinity%2C1%29-category'>(-1)-truncation</a></p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+support'>compact support</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compactly+supported+cohomology'>compactly supported cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/split+support'>split support</a></p> </li> </ul><ins class='diffins'> </ins><ins class='diffins'><h2 id='references'>References</h2></ins><ins class='diffins'> </ins><ins class='diffins'><ul> <li>Wikipedia, <em><a href='https://en.wikipedia.org/wiki/Support_(mathematics'>Support (mathematics)</a>)</em></li> </ul></ins> </div>  <div class="revisedby"> <p> Revision on April 28, 2017 at 09:53:15 by <a href="/nlab/author/Urs+Schreiber" style="color: #005c19">Urs Schreiber</a> See the <a href="/nlab/history/support" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="https://nforum.ncatlab.org/discussion/7710/#Item_6">Discuss</a><span class="backintime"><a href="/nlab/revision/diff/support/4" accesskey="F" class="navlinkbackintime" id="to_next_revision" rel="nofollow">Next revision</a> (10 more)</span><span class="backintime"><a href="/nlab/revision/diff/support/2" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a> (2 more)</span><a 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