Definition:Local Basis/Local Basis for Open Sets

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 $\BB$ 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 $\BB$.

Also see

• Equivalence of Definitions of Local Basis

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction