Topological Closure

topology

Definition

Topological Closure

Let $S$ be a subset of a topological space $X$. The closure of $S$, denoted $\overline{S}$, is the intersection of all closed sets containing $S$:

$$\overline{S}=\bigcap \{C\subseteq X\mid S\subseteq C,C\text{ closed}\}.$$

Equivalently, $\overline{S}=S\cup \{\text{limit points of }S\}$.