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Concept
Neighbourhood system
Summary
In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter for a point in a topological space is the collection of all neighbourhoods of
Neighbourhood of a point or set
An of a point (or subset) in a topological space is any open subset of that contains
A is any subset that contains open neighbourhood of ;
explicitly, is a neighbourhood of in if and only if there exists some open subset with .
Equivalently, a neighborhood of is any set that contains in its topological interior.
Importantly, a "neighbourhood" does have to be an open set; those neighbourhoods that also happen to be open sets are known as "open neighbourhoods."
Similarly, a neighbourhood that is also a closed (respectively, compact, connected, etc.) set is called a (respectively, , , etc.).
There are many other types of neighbourhoods that are used in topology and related fields like functional analysis.
The family of all neighbourhoods having a certain "useful" property often forms a neighbourhood basis, although many times, these neighbourhoods are not necessarily open. Locally compact spaces, for example, are those spaces that, at every point, have a neighbourhood basis consisting entirely of compact sets.
Neighbourhood filter
The neighbourhood system for a point (or non-empty subset) is a filter called the The neighbourhood filter for a point is the same as the neighbourhood filter of the singleton set
A or (or or ) for a point is a filter base of the neighbourhood filter; this means that it is a subset
such that for all there exists some such that
That is, for any neighbourhood we can find a neighbourhood in the neighbourhood basis that is contained in
Equivalently, is a local basis at if and only if the neighbourhood filter can be recovered from in the sense that the following equality holds:
A family is a neighbourhood basis for if and only if is a cofinal subset of with respect to the partial order (importantly, this partial order is the superset relation and not the subset relation).
Official source
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