Closure of the Set

• categories: Real Analysis, Definition

$\overline{E}=E\cup E^{\prime }$

Properties:

• $T\subseteq E\Rightarrow \overline{T}\subseteq \overline{S}$
• For S closed, $T\subseteq E\iff \overline{T}\subseteq S$

Theorem 1: $\overline{E}$ is a closed set Theorem 2: $\overline{E}$ is open $\iff$ $E^C$ is closed