Examples

Čech complex as filtered complex

Čech complex as filtered complex

Let $P = {p_{1}, \dots, p_{n}}$ be a finite set of points in a metric space $(X, d)$ .

Recall that the Čech complex of $P$ , written $P_{r}$ for $r \in [0, \infty)$ , is the nerve complex of the collection of open balls ${B_{r} (p_{i})}$ .

In other words, $P_{r}$ is the complex with vertices the points of $P$ , and $k$ -simplices the subsets $σ \subset P$ with $k + 1$ points whose $r$ -balls have nonempty mutual intersection.

This is a filtered complex because $B_{r} (p_{i}) \subset B_{s} (p_{i})$ whenever $r \leq s$ . Once in, always in, for a simplex.