Čech complex

In algebraic topology and topological data analysis, the Čech complex is an abstract simplicial complex constructed from a point cloud in any metric space which is meant to capture topological 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 ${\displaystyle {\check {C}}_{\varepsilon }(X)}$ as follows: Take the elements of X as the vertex set of ${\displaystyle {\check {C}}_{\varepsilon }(X)}$. Then, for each ${\displaystyle \sigma \subset X}$, let ${\displaystyle \sigma \in {\check {C}}_{\varepsilon }(X)}$ if the set of ε-balls centered at points of σ has a nonempty intersection. In other words, the Čech complex is the nerve of the set of ε-balls centered at points of X. By the nerve lemma, the Čech complex is homotopy equivalent to the union of the balls, also known as the offset filtration.^[1]