In algebraic topology and topological data analysis, the ÃÂech 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 õ > 0, we construct the ÃÂech complex as follows: Take the elements of X as the vertex set of . Then, for each , let if the set of õ-balls centered at points of àhas a nonempty intersection. In other words, the ÃÂech complex is the nerve of the set of õ-balls centered at points of X. By the nerve lemma, the ÃÂech complex is homotopy equivalent to the union of the balls, also known as the offset filtration.
The ÃÂech complex is a subcomplex of the VietorisâÂÂRips complex. While the ÃÂech complex is more computationally expensive than the VietorisâÂÂRips complex, since we must check for higher order intersections of the balls in the complex, the nerve theorem provides a guarantee that the ÃÂech complex is homotopy equivalent to union of the balls in the complex. The VietorisâÂÂRips complex may not be.