Closed Set

analysis topology

Definition

Closed Set

Topological Space

In a topological space, a set $S$ is closed if its complement $S^c$ is an open set.

Metric Space

In a metric space, this is equivalent to containing all limit points: for every convergent sequence $(s_n)\subseteq S$,

$$\underset{n\to \infty }{\lim }{s}_n=L⟹L\in S.$$