Union of Open Sets of Neighborhood Space is Open
Theorem
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 $\struct {S, \NN}$.
Proof
Let $\ds x \in \bigcup_{\alpha \mathop \in I} U_\alpha$.
Then $x \in U_\beta$ for some $\beta \in I$.
By definition of open set, $U_\beta$ is a neighborhood of $x$.
But from Set is Subset of Union:
- $\ds U_\beta \subseteq x \in \bigcup_{\alpha \mathop \in I} U_\alpha$
By neighborhood space axiom $N 3$ it follows that $\ds \bigcup_{\alpha \mathop \in I} U_\alpha$ is a neighborhood of $x$.
As $x$ is arbitrary, it follows that the above is true for all $\ds x \in \bigcup_{\alpha \mathop \in I} U_\alpha$.
It follows by definition that $\ds \bigcup_{\alpha \mathop \in I} U_\alpha$ is an open set of $\struct {S, \NN}$.
$\blacksquare$
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $3$: Topological Spaces: $\S 3$: Neighborhoods and Neighborhood Spaces: Lemma $3.6$