Definition:Neighborhood Space Induced by Topological Space

Theorem

Let $S$ be a set.

Let $\tau$ be a topology on $S$, thus forming the topological space $\struct {S, \tau}$.

For all $x \in S$, let $\NN_x$ be the neighborhood filter of $x$ in $S$.

Let $\NN$ be the set of subsets of $S$ such that:

$\forall N \in \NN: N$ is a neighborhood of each of its points

Then $\struct {S, \NN}$ is the neighborhood space induced by $\struct {S, \tau}$.

Also known as

Some sources refer to $\struct {S, \NN}$ in this context as the neighborhood space given rise to by $\struct {S, \tau}$.

Also see

• Induced Neighborhood Space is Neighborhood Space

Sources

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $3$: Topological Spaces: $\S 3$: Neighborhoods and Neighborhood Spaces