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Math and science::Topology

Neighbourhoods

Let X be a set with topology T. A neighbourhood of x is an open subset of X containing x.

This is the definition according to Munkres. Tom Leinster gives a different definition that distinguishes between a 'neighbourhood' and an 'open neighbourhood'.

Neighbourhood is the closest we get to the metric space idea of open balls around a point.

The term neighbourhood packs a noun-verb pair into a noun, which I think is part of why using it makes it easier to conceptualize compared to 'open set containing x'.


The next lemma is of interest because it is a lemma expressed in the terms of topology that is in some sense an isomorphism of a definition that appeared when formulating open sets in metric spaces. When defining open sets of a topology, for our definition we took the metric space's derived result of open sets being closed under arbitrary union and finite intersection. Now we have come to a derived result of topological spaces that matches the founding definition of open sets in metric spaces. Isn't that an interesting loop.

Lemma. Open iff all elements have a neighbourhood subset

Let X be a topological space and UX. Then U is open in X iff for all xU, there is a neighbourhood of x contained in U.

For comparison, the definition of open sets for a metric space is copied below:

Metric space: Open subsets, definition

Let X be a metric space and UX. Then U is open in X iff for all uU, there is a ε>0 such that B(u,ε)U.

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