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Neighbourhood system

In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filterN(x){\displaystyle {\mathcal {N}}(x)} for a point x{\displaystyle x} in a topological space is the collection of all neighbourhoods of x.{\displaystyle x.}DefinitionsNeighbourhood of a point or setAn open neighbourhood of a point (or subset) x{\displaystyle x} in a topological space X{\displaystyle X} is any open subsetU{\displaystyle U} of X{\displaystyle X} that contains x.{\displaystyle x.} A neighbourhood of x{\displaystyle x} in X{\displaystyle X} is any subset N⊆X{\displaystyle N\subseteq X} that contains some open neighbourhood of x{\displaystyle x}; explicitly, N{\displaystyle N} is a neighbourhood of x{\displaystyle x} in X{\displaystyle X}if and only if there exists some open subset U{\displaystyle U} with x∈U⊆N{\displaystyle x\in U\subseteq N}.

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