 In mathematics, an adherent point (also closure point or point of closure or contact point) of a subset $A$ of a topological space $X,$ is a point $x$ in $X$ such that every neighbourhood of $x$ (or equivalently, every open neighborhood of $x$ ) contains at least one point of $A.$ A point $x\in X$ is an adherent point for $A$ if and only if $x$ is in the closure of $A,$ thus

$x\in \operatorname {Cl} _{X}A$ if and only if for all open subsets $U\subseteq X,$ if $x\in U{\text{ then }}U\cap A\neq \varnothing .$ This definition differs from that of a limit point, in that for a limit point it is required that every neighborhood of $x$ contains at least one point of $A$ different from $x.$ Thus every limit point is an adherent point, but the converse is not true. An adherent point of $A$ is either a limit point of $A$ or an element of $A$ (or both). An adherent point which is not a limit point is an isolated point.

Intuitively, having an open set $A$ defined as the area within (but not including) some boundary, the adherent points of $A$ are those of $A$ including the boundary.

## Examples and sufficient conditions

If $S$ is a non-empty subset of $\mathbb {R}$ which is bounded above, then the supremum $\sup S$ is adherent to $S.$ In the interval $(a,b],$ $a$ is an adherent point that is not in the interval, with usual topology of $\mathbb {R} .$ A subset $S$ of a metric space $M$ contains all of its adherent points if and only if $S$ is (sequentially) closed in $M.$ Suppose $x\in X$ and $S\subseteq X\subseteq Y,$ where $X$ is a topological subspace of $Y$ (that is, $X$ is endowed with the subspace topology induced on it by $Y$ ). Then $x$ is an adherent point of $S$ in $X$ if and only if $x$ is an adherent point of $S$ in $Y.$ [proof 1] Consequently, $x$ is an adherent point of $S$ in $X$ if and only if this is true of $x$ in every (or alternatively, in some) topological superspace of $X.$ If $S$ is a subset of a topological space then the limit of a convergent sequence in $S$ does not necessarily belong to $S,$ however it is always an adherent point of $S.$ Let $\left(x_{n}\right)_{n\in \mathbb {N} }$ be such a sequence and let $x$ be its limit. Then by definition of limit, for all neighbourhoods $U$ of $x$ there exists $n\in \mathbb {N}$ such that $x_{n}\in U$ for all $n\geq N.$ In particular, $x_{N}\in U$ and also $x_{N}\in S,$ so $x$ is an adherent point of $S.$ In contrast to the previous example, the limit of a convergent sequence in $S$ is not necessarily a limit point of $S$ ; for example consider $S=\{0\}$ as a subset of $\mathbb {R} .$ Then the only sequence in $S$ is the constant sequence $0,0,\ldots$ whose limit is $0,$ but $0$ is not a limit point of $S;$ it is only an adherent point of $S.$ • Closed set – The complement of an open subset a topological space. It contains all points that are "close" to it.
• Closure (topology)
• Limit of a sequence – Value that the terms of a sequence "tend to"
• Limit point – A point x in a topological space, all of whose neighborhoods contain some other point in a given subset that is different from x
• Subsequential limit – The limit of some subsequence