In topology, the closure of a subset S of points in a topological space consists of all points in S together with all limit points of S. The closure of S may equivalently be defined as the union of S and its boundary, and also as the intersection of all closed sets containing S. Intuitively, the closure can be thought of as all the points that are either in S or "very near" S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior.

For ${\displaystyle S}$ as a subset of a Euclidean space, ${\displaystyle x}$ is a point of closure of ${\displaystyle S}$ if every open ball centered at ${\displaystyle x}$ contains a point of ${\displaystyle S}$ (this point can be ${\displaystyle x}$ itself).

This definition generalizes to any subset ${\displaystyle S}$ of a metric space ${\displaystyle X.}$ Fully expressed, for ${\displaystyle X}$ as a metric space with metric ${\displaystyle d,}$ ${\displaystyle x}$ is a point of closure of ${\displaystyle S}$ if for every ${\displaystyle r>0}$ there exists some ${\displaystyle s\in S}$ such that the distance ${\displaystyle d(x,s)<r}$ (${\displaystyle x=s}$ is allowed). Another way to express this is to say that ${\displaystyle x}$ is a point of closure of ${\displaystyle S}$ if the distance ${\displaystyle d(x,S):=\inf _{s\in S}d(x,s)=0}$ where ${\displaystyle \inf }$ is the infimum.

This definition generalizes to topological spaces by replacing "open ball" or "ball" with "neighbourhood". Let ${\displaystyle S}$ be a subset of a topological space ${\displaystyle X.}$ Then ${\displaystyle x}$ is a point of closure or adherent point of ${\displaystyle S}$ if every neighbourhood of ${\displaystyle x}$ contains a point of ${\displaystyle S}$ (again, ${\displaystyle x=s}$ for ${\displaystyle s\in S}$ is allowed). Note that this definition does not depend upon whether neighbourhoods are required to be open.

The definition of a point of closure of a set is closely related to the definition of a limit point of a set. The difference between the two definitions is subtle but important – namely, in the definition of a limit point ${\displaystyle x}$ of a set ${\displaystyle S}$, every neighbourhood of ${\displaystyle x}$ must contain a point of ${\displaystyle S}$ other than ${\displaystyle x}$ itself, i.e., each neighbourhood of ${\displaystyle x}$ obviously has ${\displaystyle x}$ but it also must have a point of ${\displaystyle S}$ that is not equal to ${\displaystyle x}$ in order for ${\displaystyle x}$ to be a limit point of ${\displaystyle S}$. A limit point of ${\displaystyle S}$ has more strict condition than a point of closure of ${\displaystyle S}$ in the definitions. The set of all limit points of a set ${\displaystyle S}$ is called the derived set of ${\displaystyle S}$. A limit point of a set is also called cluster point or accumulation point of the set.

Thus, every limit point is a point of closure, but not every point of closure is a limit point. A point of closure which is not a limit point is an isolated point. In other words, a point ${\displaystyle x}$ is an isolated point of ${\displaystyle S}$ if it is an element of ${\displaystyle S}$ and there is a neighbourhood of ${\displaystyle x}$ which contains no other points of ${\displaystyle S}$ than ${\displaystyle x}$ itself.

For a given set ${\displaystyle S}$ and point ${\displaystyle x,}$ ${\displaystyle x}$ is a point of closure of ${\displaystyle S}$ if and only if ${\displaystyle x}$ is an element of ${\displaystyle S}$ or ${\displaystyle x}$ is a limit point of ${\displaystyle S}$ (or both).

The closure of a subset ${\displaystyle S}$ of a topological space ${\displaystyle (X,\tau ),}$ denoted by ${\displaystyle \operatorname {cl} _{(X,\tau )}S}$ or possibly by ${\displaystyle \operatorname {cl} _{X}S}$ (if ${\displaystyle \tau }$ is understood), where if both ${\displaystyle X}$ and ${\displaystyle \tau }$ are clear from context then it may also be denoted by ${\displaystyle \operatorname {cl} S,}$ ${\displaystyle {\overline {S}},}$ or ${\displaystyle S{}^{-}}$ (Moreover, ${\displaystyle \operatorname {cl} }$ is sometimes capitalized to ${\displaystyle \operatorname {Cl} }$.) can be defined using any of the following equivalent definitions: