Neighbourhood system

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).