Definition:Local Basis
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 $\BB$ of open neighborhoods of $x$ such that:
- $\forall U \in \tau: x \in U \implies \exists H \in \BB: H \subseteq U$
That is, such that every open neighborhood of $x$ also contains some set in $\BB$.
Neighborhood Basis of Open Sets
A local basis at $x$ is a set $\BB$ of open neighborhoods of $x$ such that every neighborhood of $x$ contains a set in $\BB$.
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 $\BB$ 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
- Equivalence of Definitions of Local Basis
- Definition:Local Sub-Basis
- Local Basis Generated from Neighborhood Basis
- Results about local bases can be found here.