Category:Definitions/Local Bases

This category contains definitions related to Local Bases.
Related results can be found in Category:Local Bases.

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.