If \R has its usual
Euclidean topology then the neighborhoods of 0 are all those subsets N \subseteq \R for which there exists some
real number r > 0 such that (-r, r) \subseteq N. For example, all of the following sets are neighborhoods of 0 in \R: (-2, 2), \; [-2,2], \; [-2, \infty), \; [-2, 2) \cup \{10\}, \; [-2, 2] \cup \Q, \; \R but none of the following sets are neighborhoods of 0: \{0\}, \; \Q, \; (0,2), \; [0, 2), \; [0, 2) \cup \Q, \; (-2, 2) \setminus \left\{1, \tfrac{1}{2}, \tfrac{1}{3}, \tfrac{1}{4}, \ldots\right\} where \Q denotes the
rational numbers. If U is an open subset of a
topological space X then for every u \in U, U is a neighborhood of u in X. More generally, if N \subseteq X is any set and \operatorname{int}_X N denotes the
topological interior of N in X, then N is a neighborhood (in X) of every point x \in \operatorname{int}_X N and moreover, N is a neighborhood of any other point. Said differently, N is a neighborhood of a point x \in X if and only if x \in \operatorname{int}_X N.
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 x in a
metric space, the sequence of
open balls around x with radius 1/n form a
countable neighbourhood basis \mathcal{B} = \left\{B_{1/n} : n = 1,2,3,\dots \right\}. This means every metric space is
first-countable. Given a space X with the
indiscrete topology the neighbourhood system for any point x only contains the whole space, \mathcal{N}(x) = \{X\}. In the
weak topology on the space of measures on a space E, a neighbourhood base about \nu is given by \left\{\mu \in \mathcal{M}(E) : \left|\mu f_i - \nu f_i\right| where f_i are
continuous bounded functions from E to the real numbers and r_1, \dots, r_n 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, \mathcal{N}(x) = \mathcal{N}(0) + x. 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==