Neighbourhood basis

Given a topological space $(𝑋, T)$ , a neighbourhood basis $B (𝑥) \subseteq N (𝑥)$ of a point $𝑥 \in 𝑋$ is a set of neighbourhoods of $𝑥$ such that given any neighbourhood $𝑆 \in N (𝑥)$ , there exists a basic neighbourhood $𝐵 \in B (𝑥)$ such that $𝑥 \in 𝐵 \subseteq 𝑆 \subseteq 𝑋$ . #m/def/topology

Open neighbourhood basis

Note that since every neighbourhood contains an open neighbourhood, we can always give an open neighbourhood basis if we have a neighbourhood basis, without loss of generality.

Properties

A topological space is first-countable iff every point has a countable neighbourhood basis.
Given a countable neighbourhood basis, it is possible to construct a Nested neighbourhood basis.

#state/tidy | #lang/en | #SemBr