Čech complex

In <a href="/facts/Algebraic_topology/2wA8yrjT">algebraic topology</a> and <a href="/facts/Topological_data_analysis/Lz5ltT7l">topological data analysis</a>, the Čech complex is an <a href="/facts/Abstract_simplicial_complex/ytwxClsf">abstract simplicial complex</a> constructed from a point cloud in any <a href="/facts/Metric_space/gnBBRXtc">metric space</a> which is meant to capture <a href="/facts/Topology/2EqW47cU">topological</a> information about the point cloud or the distribution it is drawn from. Given a finite point cloud X and an ε > 0, we construct the Čech complex 
 
 
 
 
 
 
 
 C
 ˇ
 
 
 
 
 ε
 
 
 (
 X
 )
 
 
 {\displaystyle {\check {C}}_{\varepsilon }(X)}
 
 as follows: Take the elements of X as the vertex set of 
 
 
 
 
 
 
 
 C
 ˇ
 
 
 
 
 ε
 
 
 (
 X
 )
 
 
 {\displaystyle {\check {C}}_{\varepsilon }(X)}
 
. Then, for each 
 
 
 
 σ
 ⊂
 X
 
 
 {\displaystyle \sigma \subset X}
 
, let 
 
 
 
 σ
 ∈
 
 
 
 
 C
 ˇ
 
 
 
 
 ε
 
 
 (
 X
 )
 
 
 {\displaystyle \sigma \in {\check {C}}_{\varepsilon }(X)}
 
 if the set of ε-balls centered at points of σ has a <a href="/facts/Empty_set/ffEk3eA6">nonempty</a> <a href="/facts/Intersection_(set_theory)/RPfUNJh9">intersection</a>. In other words, the Čech complex is the <a href="/facts/Nerve_of_a_covering/oeV3qb6f">nerve</a> of the set of ε-balls centered at points of X. By the <a href="/facts/Nerve_of_a_covering/oeV3qb6f">nerve lemma</a>, the Čech complex is homotopy equivalent to the union of the balls, also known as the <a href="/facts/Offset_filtration/CMxP4xoC">offset filtration</a>.

Čech complex open-in-new

Čech complex