Neighbourhood system

Neighbourhood basis

Given a topological space , a neighbourhood basis of a point is a set of neighbourhoods of such that given any neighbourhood , there exists a basic neighbourhood such that . #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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