# Union of Open Sets is Open

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## 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 $\family {U_\alpha}_{\alpha \mathop \in I}$ be a family of open sets of $\struct {S, \NN}$ indexed by $I$.

Then their union $\ds \bigcup_{\alpha \mathop \in I} U_i$ is an open set of $\struct {S, \NN}$.