Neighbourhood system

Neighbourhood basis

Given a topological space (𝑋,T), a neighbourhood basis B(𝑥) ⊆N(𝑥) of a point 𝑥 ∈𝑋 is a set of neighbourhoods of 𝑥 such that given any neighbourhood 𝑆 ∈N(𝑥), there exists a basic neighbourhood 𝐵 ∈B(𝑥) such that 𝑥 ∈𝐵 ⊆𝑆 ⊆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.

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