Definition:Neighborhood Basis
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Definition
Let $\struct {S, \tau}$ be a topological space.
Let $x \in S$.
Let $\BB$ be a set of neighborhoods of $x$.
Then $\BB$ is a neighborhood basis at $x$ if and only if:
- For each neighborhood $N$ of $x$, there is an $M \in \BB$ such that $M \subseteq N$.
Also defined as
Some sources require the elements of a neighborhood basis to be open.
On $\mathsf{Pr} \infty \mathsf{fWiki}$, such a structure is referred to as a local basis.
Also known as
Some sources call this a local basis, but on $\mathsf{Pr} \infty \mathsf{fWiki}$, that term is reserved for a stronger notion.
Also see
- Results about neighborhood bases can be found here.
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $3$: Topological Spaces: $\S 3$: Neighborhoods and Neighborhood Spaces: Exercise $3$