Definition:Local Basis

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Definition

Let $T = \struct {S, \tau}$ be a topological space.

Let $x$ be an element of $S$.


Local Basis for Open Sets

A local basis at $x$ is a set $\mathcal B$ of open neighborhoods of $x$ such that:

$\forall U \in \tau: x \in U \implies \exists H \in \mathcal B: H \subseteq U$

That is, such that every open neighborhood of $x$ also contains some set in $\mathcal B$.


Neighborhood Basis of Open Sets

A local basis at $x$ is a set $\mathcal B$ of open neighborhoods of $x$ such that every neighborhood of $x$ contains a set in $\mathcal B$.

That is, a local basis at $x$ is a neighborhood basis of $x$ consisting of open sets.


Also defined as

Some more modern sources suggest that in order to be a local basis, the neighborhoods of which the set $\mathcal B$ consists do not need to be open.

Such a structure is referred to on $\mathsf{Pr} \infty \mathsf{fWiki}$ as a neighborhood basis.


Also known as

A local basis is also known as a neighborhood basis, but that term is used on $\mathsf{Pr} \infty \mathsf{fWiki}$ for a weaker notion.


Also see