Adherent point

In mathematics, an adherent point (also closure point or point of closure or contact point)^[1] of a subset ${\displaystyle A}$ of a topological space ${\displaystyle X,}$ is a point ${\displaystyle x}$ in ${\displaystyle X}$ such that every neighbourhood of ${\displaystyle x}$ (or equivalently, every open neighborhood of ${\displaystyle x}$) contains at least one point of ${\displaystyle A.}$ A point ${\displaystyle x\in X}$ is an adherent point for ${\displaystyle A}$ if and only if ${\displaystyle x}$ is in the closure of ${\displaystyle A,}$ thus

${\displaystyle x\in \operatorname {Cl} _{X}A}$ if and only if for all open subsets ${\displaystyle U\subseteq X,}$ if ${\displaystyle x\in U{\text{ then }}U\cap A\neq \varnothing .}$

This definition differs from that of a limit point of a set, in that for a limit point it is required that every neighborhood of ${\displaystyle x}$ contains at least one point of ${\displaystyle A}$ different from ${\displaystyle x.}$ Thus every limit point is an adherent point, but the converse is not true. An adherent point of ${\displaystyle A}$ is either a limit point of ${\displaystyle A}$ or an element of ${\displaystyle A}$ (or both). An adherent point which is not a limit point is an isolated point.

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