in a topological space is the collection of all neighbourhoods of
Neighbourhood of a point or set An open neighbourhood of a point (or subset[note 1])
that contains some open neighbourhood of
Importantly, a "neighbourhood" does not have to be an open set; those neighbourhoods that also happen to be open sets are known as "open neighbourhoods.
"[note 2] Similarly, a neighbourhood that is also a closed (respectively, compact, connected, etc.)
There are many other types of neighbourhoods that are used in topology and related fields like functional analysis.
Locally compact spaces, for example, are those spaces that, at every point, have a neighbourhood basis consisting entirely of compact sets.
The neighbourhood filter for a point
is the same as the neighbourhood filter of the singleton set
A neighbourhood basis or local basis (or neighbourhood base or local base) for a point
denotes the set of all neighbourhoods of x.
in the neighbourhood basis that is contained in
with respect to the partial order
such that the collection of all possible finite intersections of elements of
forms a neighbourhood basis at
has its usual Euclidean topology then the neighborhoods of
denotes the rational numbers.
is an open subset of a topological space
The set of all open neighbourhoods at a point forms a neighbourhood basis at that point.
in a metric space, the sequence of open balls around
form a countable neighbourhood basis
This means every metric space is first-countable.
with the indiscrete topology the neighbourhood system for any point
are continuous bounded functions from
are positive real numbers.
Seminormed spaces and topological groups In a seminormed space, that is a vector space with the topology induced by a seminorm, all neighbourhood systems can be constructed by translation of the neighbourhood system for the origin,
This is because, by assumption, vector addition is separately continuous in the induced topology.
Therefore, the topology is determined by its neighbourhood system at the origin.
More generally, this remains true whenever the space is a topological group or the topology is defined by a pseudometric.
into a directed set by partially ordering it by superset inclusion