nLab neighborhood base

Definition

Let $(X, \tau)$ be a topological space, and $x \in X$ a point. A neighborhood base at $x$ is a collection $\{U_i \subset X\}_{i \in I}$ of neighbourhoods of $x$ such that every neighborhood $W$ of $x$ (which WLOG we may assume open) contains some $U_i$:

$$\underset{ { W \underset{\text{open}}{\subset} X } \atop { W \supset \{x\} } }{\forall} \left( \underset{i \in I}{\exists} \left( U_i \subset W \right) \right) \,.$$

Related concepts

• base of a topology

• locally compact topological space

References

See also

• Wikipeda, Neighbourhood system – Basis