In topology, the closure of a subset (A) of a topological space (X) is the smallest closed set in (X) containing (A). It is commonly written (\overline{A}), (\operatorname{cl}(A)), or (\operatorname{cl}_X(A)). Closure includes the original set together with all points that cannot be separated from it by an open neighborhood. The ambient space and its topology are essential: the same set can have different closures in different spaces. (maths.ed.ac.uk)
Definition and neighborhood characterization
Formally, [ \operatorname{cl}_X(A)
\bigcap{F\subseteq X: F\text{ is closed in }X,\ A\subseteq F}. ] The family being intersected is nonempty because it contains (X). Arbitrary intersections of closed sets are closed, so this expression is itself closed and establishes the existence of the smallest closed superset. In particular, (A) is closed exactly when (\operatorname{cl}_X(A)=A). (sites.math.northwestern.edu)
An equivalent characterization uses neighborhoods: [ x\in\operatorname{cl}_X(A) \quad\Longleftrightarrow\quad U\cap A\ne\varnothing \text{ for every open neighborhood }U\text{ of }x. ] Thus, (x) lies outside the closure precisely when some open set containing (x) misses (A). Points satisfying the neighborhood condition are called adherent points or closure points of (A). (maths.ed.ac.uk)
Closure points should be distinguished from accumulation points, for which every neighborhood must meet (A\setminus{x}). If (A') denotes the set of accumulation points, then [ \overline{A}=A\cup A'. ] Consequently, an isolated point of (A) belongs to its closure even though it is not an accumulation point. No Hausdorff assumption is needed for these characterizations. (sites.math.northwestern.edu)
Closure as an operator
Closure defines a map from the power set of (X) to itself. Its fundamental properties are the Kuratowski closure axioms: [ \begin{aligned} \operatorname{cl}(\varnothing)&=\varnothing,\ A&\subseteq\operatorname{cl}(A),\ \operatorname{cl}(\operatorname{cl}(A))&=\operatorname{cl}(A),\ \operatorname{cl}(A\cup B)&=\operatorname{cl}(A)\cup\operatorname{cl}(B). \end{aligned} ] They express preservation of the empty set, extensivity, idempotence, and preservation of finite unions. Conversely, any operator on the subsets of (X) satisfying these axioms determines a unique topology: its closed sets are exactly the subsets (F) for which (\operatorname{cl}(F)=F). Their complements are the open sets. (legacy-www.math.harvard.edu)
Closure is also monotone: (A\subseteq B) implies (\overline{A}\subseteq\overline{B}). However, it need not preserve intersections or infinite unions. For example, in the usual real line, (A=(0,1)) and (B=(1,2)) have empty intersection, while their closures intersect at (1). Likewise, the union of the closed singletons ({1/n}), for positive integers (n), is not closed: its closure additionally contains (0). These examples illustrate why the finite-union axiom cannot be extended to arbitrary unions. (fan.uni-wuppertal.de)
Metric spaces and convergence
In a metric space ((X,d)), the neighborhood criterion becomes [ x\in\overline{A} \quad\Longleftrightarrow\quad B(x,\varepsilon)\cap A\ne\varnothing \quad\text{for every }\varepsilon>0, ] where (B(x,\varepsilon)) is an open ball. For nonempty (A), this is equivalent to [ d(x,A):=\inf_{a\in A}d(x,a)=0. ] The infimum need not be attained: a point outside (A) may nevertheless have distance zero from it. (maths.ed.ac.uk)
In metric spaces, and more generally in every first-countable space, a point belongs to (\overline{A}) exactly when it is the limit of a sequence of points of (A). In the metric case, one can choose (a_n\in A\cap B(x,1/n)); then (a_n\to x). Points already in (A) are covered by constant sequences. (math.ucla.edu)
In arbitrary topological spaces, sequential limits may capture only part of the closure. The unrestricted characterization uses a net, a generalization of a sequence indexed by a directed set: (x\in\overline{A}) if and only if some net taking values in (A) converges to (x). (fan.uni-wuppertal.de)
Examples and dependence on topology
With the usual topology on the real numbers, [ \overline{(0,1)}=[0,1], \qquad \overline{\mathbb Q}=\mathbb R, \qquad \overline{\mathbb R\setminus\mathbb Q}=\mathbb R. ] Both the rational numbers and the irrational numbers are therefore dense in (\mathbb R). Generally, a subset is dense in (X) exactly when its closure is (X). (maths.ed.ac.uk)
In the discrete topology, every subset is closed and equals its closure. In the indiscrete topology, the only closed subsets are (\varnothing) and (X), so every nonempty subset has closure (X). These cases show that closure describes topological proximity rather than necessarily numerical distance. (math.mit.edu)
If (A\subseteq Y\subseteq X), with (Y) carrying the subspace topology, then [ \operatorname{cl}_Y(A)=Y\cap\operatorname{cl}_X(A). ] For instance, ((0,1)) has closure ([0,1]) in (\mathbb R), but closure ((0,1]) in (Y=(0,2)). (maths.ed.ac.uk)
Interior and continuous maps
Closure is dual to interior under complementation: [ \operatorname{cl}_X(A)
X\setminus\operatorname{int}_X(X\setminus A). ] A continuous map (f:X\to Y) satisfies [ f(\operatorname{cl}_X(A)) \subseteq \operatorname{cl}_Y(f(A)) ] for every (A\subseteq X); conversely, this condition characterizes continuity. Equality is not required. For example, the continuous map (f(x)=e^x) sends the closed set (\mathbb R) onto ((0,\infty)), whose closure in (\mathbb R) also contains (0). (fan.uni-wuppertal.de)