Definition:Local Basis/Local Basis for Open Sets

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Definition

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

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


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 $\mathcal B$.


Also see


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