Neighborhood (topology)

In topology, a neighborhood of a point is a set containing an open set that contains that point. It describes the surroundings of a point in a topological space without requiring a numerical distance. Neighborhoods provide a language for defining continuity, convergence, and other local properties. A neighborhood need not itself be open; one that is open is called an open neighborhood. Some authors use “neighborhood” only for open neighborhoods, so the convention matters. (pi.math.cornell.edu)

Definition and examples

Let $(X,\tau)$ be a topological space and $x\in X$. A subset $N\subseteq X$ is a neighborhood of $x$ if there exists an open set $U\in\tau$ such that

Equivalently, $x$ belongs to the interior of $N$. Thus, containing $x$ is necessary but not sufficient: the set must contain an open region around $x$. A set is open precisely when it is a neighborhood of every point it contains. (math.berkeley.edu)

In the usual topology on the real numbers, both $(-1,1)$ and $[-1,1]$ are neighborhoods of $0$. The latter is a closed set, illustrating that neighborhoods need not be open. However, $[0,1]$ is not a neighborhood of $0$ in $\mathbb R$, since no open interval containing $0$ lies inside it. These examples follow directly from the definition. (pi.math.cornell.edu)

In a metric space $(X,d)$, the definition becomes

where the open ball is

For Euclidean space, these are the familiar intervals, disks, and higher-dimensional balls. Neighborhoods may also contain additional, distant points; they need not resemble balls geometrically. (pi.math.cornell.edu)

Neighborhood systems and axioms

The collection of all neighborhoods of $x$, denoted $\mathcal N(x)$, is its neighborhood system. It is a filter: it is nonempty, excludes the empty set, is closed under finite intersections, and contains every superset of each of its members. Every member contains $x$. Arbitrary intersections need not remain neighborhoods; for example,

is not a neighborhood of $0$ in $\mathbb R$. (math.berkeley.edu)

Neighborhood systems can define a topology directly through axioms. In addition to the preceding requirements, they satisfy a local compatibility condition: for every $N\in\mathcal N(x)$, there is $M\in\mathcal N(x)$, with $M\subseteq N$, such that $N\in\mathcal N(y)$ for every $y\in M$. The open sets are then exactly those sets $O$ satisfying $O\in\mathcal N(x)$ for every $x\in O$. This reconstruction uniquely determines the topology. (math.berkeley.edu)

Neighborhood bases

A neighborhood base, or local base, at $x$ is a collection $\mathcal B_x\subseteq\mathcal N(x)$ such that every neighborhood of $x$ contains some member of $\mathcal B_x$. It supplies enough neighborhoods to test local conditions without considering the entire neighborhood system. Its members may be chosen open, although openness is not required. (fan.uni-wuppertal.de)

A basis for a topology supplies a local base at each point: take all basis elements containing that point. The distinction is that a topological basis describes open sets throughout the space, whereas a neighborhood base describes the surroundings of one point. (math.berkeley.edu)

A first-countable space has a local base with at most countably many members at every point. Every metric space has this property, since

is a local base at $x$. First countability is weaker than having a countable basis for the whole topology, the property called second countability. (fan.uni-wuppertal.de)

Continuity, convergence, and closure

A function $f:X\to Y$ is continuous at $x$ precisely when, for every neighborhood $V$ of $f(x)$, there is a neighborhood $U$ of $x$ with

Equivalently, $f^{-1}(V)$ is a neighborhood of $x$. It suffices to test this condition using local bases. (fan.uni-wuppertal.de)

A sequence $(x_n)$ has limit $x$ if every neighborhood of $x$ contains all sufficiently late terms. The same definition applies to a net, replacing natural-number indices with a directed index set. Nets allow convergence to describe arbitrary topologies, where sequences alone may be insufficient. (math.berkeley.edu)

For $A\subseteq X$, a point $x$ lies in its closure exactly when every neighborhood of $x$ intersects $A$. In first-countable spaces, this is equivalent to the existence of a sequence in $A$ converging to $x$. Likewise, for functions with first-countable domains, preserving limits of sequences characterizes continuity. Neither sequential characterization holds for all topological spaces. (sites.math.northwestern.edu)

Dependence on the ambient space

Neighborhoods are relative to a topology and an ambient space. If $Y\subseteq X$ carries the subspace topology, neighborhoods of $y\in Y$ are precisely sets $Y\cap N$, where $N$ is a neighborhood of $y$ in $X$. Consequently, $[0,\varepsilon)$ is an open neighborhood of $0$ in $[0,1]$ for $0<\varepsilon\leq1$, although it is not a neighborhood of $0$ in $\mathbb R$. (pi.math.cornell.edu)

Neighborhoods also express separation properties. A Hausdorff space is one in which any two distinct points have disjoint neighborhoods. This condition ensures that a convergent sequence or net cannot have two distinct limits. (math.ucla.edu)