Open_set

Definition

A set $O \subseteq \mathbb{R}$ is open if for all points $a \in O$ there exists an $\epsilon$-neighborhood (see epsilon-neighborhood) $V_{\epsilon}\left. (a) \right. \subseteq O$.

Theorems

(i)

The union of an arbitrary collection of open sets is open.

Proof: see here.

(ii)

The interinterion of a finite collection of open sets is open.

Proof: see here.

(iii)

$A$ set $O$ is open if and only if $O^{c}$ is a closed_set. Likewise, $F$ is a closed_set if and only if $F^{c}$ is open.