such that For example, all of the following sets are neighborhoods of in : but none of the following sets are neighborhoods of : where denotes the rational numbers. If is an open subset of a topological space then for every is a neighborhood of in More generally, if is any set and denotes the topological interior of in then is a neighborhood (in ) of every point and moreover, is a neighborhood of any other point. Said differently, is a neighborhood of a point if and only if Neighbourhood bases In any topological space, the neighbourhood system for a point is also a neighbourhood basis for the point. The set of all open neighbourhoods at a point forms a neighbourhood basis at that point. For any point in a metric space, the sequence of open balls around with radius form a countable neighbourhood basis . This means every metric space is first-countable. Given a space with the indiscrete topology the neighbourhood system for any point only contains the whole space, . In the weak topology on the space of measures on a space a neighbourhood base about is given by where are continuous bounded functions from to the real numbers and 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. Properties Suppose and let be a neighbourhood basis for in Make into a directed set by partially ordering it by superset inclusion Then is a neighborhood of in if and only if there exists an -indexed net in such that for every (which implies that in ). See also References Bibliography General topology