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Čech complex
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<h2 id="relation-to-vietorisrips-complex">Relation to Vietoris–Rips complex</h2> <p>The Čech complex is a subcomplex of the <a href="/facts/Vietoris%25E2%2580%2593Rips_complex/Ex4Px4wC">Vietoris–Rips complex</a>. While the Čech complex is more computationally expensive than the Vietoris–Rips complex, since we must check for higher order intersections of the balls in the complex, the nerve theorem provides a guarantee that the Čech complex is homotopy equivalent to union of the balls in the complex. The Vietoris–Rips complex may not be.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/%25C4%258Cech_cohomology/HLWD7BnA">Čech cohomology</a></li> <li><a href="/facts/Computational_geometry/eeaotQtl">Computational geometry</a></li> <li><a href="/facts/Simplicial_complex/fF2282CF">Simplicial complex</a></li> <li><a href="/facts/Simplicial_homology/Tc3I3IJh">Simplicial homology</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Ghrist, Robert W. (2014). Elementary applied topology (1st ed.). [United States]. ISBN 9781502880857. OCLC 899283974.{{cite book}}: CS1 maint: location missing publisher (link) <a href="9781502880857" target="_blank">9781502880857</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Ghrist, Robert W. (2014). Elementary applied topology (1st ed.). [United States]. ISBN 9781502880857. OCLC 899283974.{{cite book}}: CS1 maint: location missing publisher (link) <a href="9781502880857" target="_blank">9781502880857</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>