Union of Open Sets is Open

Theorem

Metric Space

Let $M = \struct {A, d}$ be a metric space.

The union of a set of open sets of $M$ is open in $M$.

Normed Vector Space

Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space.

The union of a set of open sets of $M$ is open in $M$.

Neighborhood Space

Let $S$ be a neighborhood space.

Let $I$ be an indexing set.

Let $\left \langle{U_\alpha}\right \rangle_{\alpha \mathop \in I}$ be a family of open sets of $\left({S, \mathcal N}\right)$ indexed by $I$.

Then their union $\displaystyle \bigcup_{\alpha \mathop \in I} U_i$ is an open set of $\left({S, \mathcal N}\right)$.