Neighbourhood (mathematics)

A neighbourhood of a point is a set that contains an open set containing that point. Neighbourhoods express the local structure of a topological space without requiring a notion of distance. They provide intrinsic formulations of continuity, convergence, closure, and other properties determined by behaviour near individual points.

Let (X) be a topological space with topology (\tau), and let (x\in X). A subset (N\subseteq X) is a neighbourhood of (x) when there exists (U\in\tau) such that

[ x\in U\subseteq N. ]

Under this definition, a neighbourhood need not itself be open. An open neighbourhood of (x) is an open set containing (x). The family of all neighbourhoods of (x) is commonly denoted by (\mathcal N(x)).

Elementary characterizations

A subset (U\subseteq X) is open precisely when it is a neighbourhood of each of its points. Consequently, the topology can be reconstructed from the neighbourhood families by the formula

[ \tau={U\subseteq X : U\in\mathcal N(x)\text{ for every }x\in U}. ]

The interior of a set (A\subseteq X) consists of the points for which (A) is a neighbourhood:

[ \operatorname{int}(A)={x\in X:A\in\mathcal N(x)}. ]

A point (x) belongs to the closure of (A) exactly when every neighbourhood of (x) intersects (A). Thus,

[ x\in\overline A \quad\Longleftrightarrow\quad N\cap A\ne\varnothing \text{ for every }N\in\mathcal N(x). ]

This criterion includes points of (A) automatically because every neighbourhood of (x) contains (x). The corresponding characterization of the boundary requires every neighbourhood of (x) to meet both (A) and its complement:

[ x\in\partial A \quad\Longleftrightarrow\quad N\cap A\ne\varnothing \ \text{and}
N\cap(X\setminus A)\ne\varnothing \text{ for every }N\in\mathcal N(x). ]

These formulas show that neighbourhoods encode the standard derived-set operations of topology.

Neighbourhood systems

Topology can be axiomatized directly through a neighbourhood assignment. For each (x\in X), let (\mathcal N(x)) be a collection of subsets of (X). Such an assignment determines a topology when it satisfies the following conditions.

  1. Every member of (\mathcal N(x)) contains (x).
  2. Every superset of a member of (\mathcal N(x)) also belongs to (\mathcal N(x)).
  3. The intersection of two members of (\mathcal N(x)) belongs to (\mathcal N(x)).
  4. For every (N\in\mathcal N(x)), there is an (M\in\mathcal N(x)) such that (N\in\mathcal N(y)) for every (y\in M).

The fourth condition expresses local stability: a neighbourhood of (x) remains a neighbourhood at every point in some sufficiently small neighbourhood of (x). The open sets reconstructed from these families form a topology, and the reconstructed neighbourhoods are exactly the original families.

The axiomatic treatment developed from the early twentieth-century separation of topology from metric geometry. In 1934, You Watanabe formulated the neighbourhood–interior correspondence in operator form, identifying the above local axioms with an interior operator satisfying extensivity in the reverse inclusion order, finite-intersection preservation, idempotence, and preservation of the whole space. Her formulation established that the assignments

[ \operatorname{int}(A)={x:A\in\mathcal N(x)} ]

and

[ \mathcal N(x)={A:x\in\operatorname{int}(A)} ]

are mutually inverse. This correspondence is now one of the standard equivalences among topological, neighbourhood, closure, and interior axiomatizations.

Neighbourhood bases

A neighbourhood basis at (x) is a subfamily (\mathcal B(x)\subseteq\mathcal N(x)) such that every neighbourhood of (x) contains some member of (\mathcal B(x)). Equivalently,

[ N\in\mathcal N(x) \quad\Longleftrightarrow\quad B\subseteq N \text{ for some }B\in\mathcal B(x). ]

A neighbourhood basis therefore records all local information while omitting sets obtained merely by enlargement. Its members need not be open, although every neighbourhood basis can be replaced by one consisting of open sets.

The local intersection property requires that for any (B_1,B_2\in\mathcal B(x)), there exists (B_3\in\mathcal B(x)) satisfying

[ B_3\subseteq B_1\cap B_2. ]

Compatibility between different points requires that whenever (y\in B\in\mathcal B(x)), there is a basis neighbourhood of (y) contained in (B), after (B) has been replaced by an appropriate local refinement when necessary. These conditions reproduce the stability and finite-intersection properties of complete neighbourhood systems.

A space is first-countable when every point has a countable neighbourhood basis. It is second-countable when the entire topology has a countable basis. The first condition is local, whereas the second imposes a countability restriction simultaneously across the whole space.

Metric neighbourhoods

In a metric space ((X,d)), the open ball with centre (x) and radius (r>0) is

[ B_r(x)={y\in X:d(x,y)<r}. ]

The family of all positive-radius open balls centred at (x) is a neighbourhood basis at (x). Consequently, a set (N) is a neighbourhood of (x) exactly when there is an (r>0) such that

[ B_r(x)\subseteq N. ]

Metric balls provide a concrete realization of neighbourhoods, but their numerical radii are not topological invariants. Two metrics induce the same topology precisely when they determine the same neighbourhood systems at every point.

For the usual topology on (\mathbb R), every open interval containing (x) is an open neighbourhood of (x). A half-closed interval such as ([x-\varepsilon,x+\varepsilon)) is also a neighbourhood because it contains a smaller open interval around (x), despite not being open itself.

A punctured neighbourhood of (x) is obtained by deleting (x) from a neighbourhood. In a metric space, the standard punctured ball is

[ B_r(x)\setminus{x}

{y\in X:0<d(x,y)<r}. ]

Punctured neighbourhoods occur in definitions concerning limiting behaviour away from the value at the distinguished point. They are not generally neighbourhoods of (x), since they do not contain (x).

Convergence and continuity

A net ((x_\alpha)) converges to (x) when it is eventually contained in every neighbourhood of (x):

[ x_\alpha\longrightarrow x \quad\Longleftrightarrow\quad \forall N\in\mathcal N(x),\ \exists\alpha_0\ \forall\alpha\geq\alpha_0,\ x_\alpha\in N. ]

This definition characterizes the topology without countability assumptions. A subset (A\subseteq X) is closed precisely when every convergent net whose terms lie in (A) has its limit in (A).

In first-countable spaces, sequences are sufficient for detecting closure and continuity because a countable local basis can be arranged into a decreasing sequence of neighbourhoods. In arbitrary topological spaces, sequences do not necessarily encode the complete neighbourhood structure. This distinction motivated the systematic use of nets by E. H. Moore and Herman L. Smith in their treatment of generalized convergence.

A function (f:X\to Y) is continuous at (x\in X) exactly when the inverse image of every neighbourhood of (f(x)) is a neighbourhood of (x):

[ V\in\mathcal N(f(x)) \quad\Longrightarrow\quad f^{-1}(V)\in\mathcal N(x). ]

Equivalently, for every neighbourhood (V) of (f(x)), there exists a neighbourhood (U) of (x) satisfying

[ f(U)\subseteq V. ]

This local formulation is equivalent to the global requirement that inverse images of open sets be open. It also extends directly to continuity expressed through convergence of nets.

Filters and local structure

For each point (x), the family (\mathcal N(x)) is a filter on (X). It is upward closed under inclusion, closed under finite intersections, and excludes the empty set. This filter is called the neighbourhood filter of (x).

Henri Cartan’s formulation of filters in the 1930s placed neighbourhood systems within a general theory of convergence. A filter (\mathcal F) converges to (x) when it contains the neighbourhood filter:

[ \mathcal N(x)\subseteq\mathcal F. ]

The corresponding relationship for an ultrafilter yields a characterization of compactness: a topological space is compact exactly when every ultrafilter on the space converges to at least one point. In a Hausdorff space, a convergent filter has at most one limit whenever the filter is an ultrafilter or otherwise satisfies the relevant convergence conditions.

Felix Hausdorff’s axiomatization of topological separation used neighbourhoods to express when distinct points can be distinguished locally. The Hausdorff condition requires that any two distinct points (x) and (y) possess disjoint neighbourhoods. In symbols,

[ x\ne y \quad\Longrightarrow\quad \exists U\in\mathcal N(x)\ \exists V\in\mathcal N(y) \text{ such that }U\cap V=\varnothing. ]

Other separation axioms alter the required relationship among points, closed sets, and their neighbourhoods.

Neighbourhoods of subsets

For a subset (A\subseteq X), a set (N\subseteq X) is a neighbourhood of (A) when it contains an open set that contains all of (A). Thus,

[ N\text{ is a neighbourhood of }A \quad\Longleftrightarrow\quad \exists U\in\tau\text{ with }A\subseteq U\subseteq N. ]

This definition is stronger than requiring (N) to be a neighbourhood of one point of (A), but it is equivalent to requiring that (N) be a neighbourhood of every point of (A). The equivalence follows because the union of the individual open subsets witnessing the pointwise condition is an open set containing (A).

A closed neighbourhood of (A) is a closed set that is also a neighbourhood of (A). The existence of appropriately nested closed neighbourhoods is central to regular spaces and normal spaces. In a regular space, each point and its neighbourhood admit a smaller neighbourhood whose closure lies inside the original neighbourhood:

[ x\in U,\ U\text{ open} \quad\Longrightarrow\quad \exists V\text{ open such that } x\in V\subseteq\overline V\subseteq U. ]

This property relates local neighbourhood refinement to the separation of points from closed sets.

See also