Closed Set

analysis topology

Definition

Closed Set

A set $S$ in a topological space is called closed if its complement ($S^c$) is an open set.

In metric spaces, a set $S$ is closed if and only if it contains all of its limit points. That is, for every convergent sequence $(s_n{)}_{n\in \mathbb{N}}\subseteq S$, the limit also belongs to $S$:

$$\forall (s\ast n)\subseteq S:\left(\lim \ast n\to \infty s_n=L⟹L\in S\right)$$

Properties

• Closure: The closure $\bar{S}$ of any set $S$ is the smallest closed set containing $S$. A set is closed if and only if $S=\bar{S}$.
• Intersections and Unions:
  • The intersection of any collection of closed sets is closed.
  • The union of a finite collection of closed sets is closed.
• Continuity: A function $f$ is continuous if and only if the preimage of every closed set is closed.
• Compactness: In Euclidean space, a set is compact if and only if it is closed and bounded.