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.

If ${\displaystyle S}$ is a non-empty subset of ${\displaystyle \mathbb {R} }$ which is bounded above, then the supremum ${\displaystyle \sup S}$ is adherent to ${\displaystyle S.}$ In the interval ${\displaystyle (a,b],}$ ${\displaystyle a}$ is an adherent point that is not in the interval, with usual topology of ${\displaystyle \mathbb {R} .}$

A subset ${\displaystyle S}$ of a metric space ${\displaystyle M}$ contains all of its adherent points if and only if ${\displaystyle S}$ is (sequentially) closed in ${\displaystyle M.}$

Suppose ${\displaystyle x\in X}$ and ${\displaystyle S\subseteq X\subseteq Y,}$ where ${\displaystyle X}$ is a topological subspace of ${\displaystyle Y}$ (that is, ${\displaystyle X}$ is endowed with the subspace topology induced on it by ${\displaystyle Y}$). Then ${\displaystyle x}$ is an adherent point of ${\displaystyle S}$ in ${\displaystyle X}$ if and only if ${\displaystyle x}$ is an adherent point of ${\displaystyle S}$ in ${\displaystyle Y.}$

Consequently, ${\displaystyle x}$ is an adherent point of ${\displaystyle S}$ in ${\displaystyle X}$ if and only if this is true of ${\displaystyle x}$ in every (or alternatively, in some) topological superspace of ${\displaystyle X.}$

If ${\displaystyle S}$ is a subset of a topological space then the limit of a convergent sequence in ${\displaystyle S}$ does not necessarily belong to ${\displaystyle S,}$ however it is always an adherent point of ${\displaystyle S.}$ Let ${\displaystyle \left(x_{n}\right)_{n\in \mathbb {N} }}$ be such a sequence and let ${\displaystyle x}$ be its limit. Then by definition of limit, for all neighbourhoods ${\displaystyle U}$ of ${\displaystyle x}$ there exists ${\displaystyle n\in \mathbb {N} }$ such that ${\displaystyle x_{n}\in U}$ for all ${\displaystyle n\geq N.}$ In particular, ${\displaystyle x_{N}\in U}$ and also ${\displaystyle x_{N}\in S,}$ so ${\displaystyle x}$ is an adherent point of ${\displaystyle S.}$ In contrast to the previous example, the limit of a convergent sequence in ${\displaystyle S}$ is not necessarily a limit point of ${\displaystyle S}$; for example consider ${\displaystyle S=\{0\}}$ as a subset of ${\displaystyle \mathbb {R} .}$ Then the only sequence in ${\displaystyle S}$ is the constant sequence ${\displaystyle 0,0,\ldots }$ whose limit is ${\displaystyle 0,}$ but ${\displaystyle 0}$ is not a limit point of ${\displaystyle S;}$ it is only an adherent point of ${\displaystyle S.}$

• Closed set – Complement of an open subset
• Closure (topology) – All points and limit points in a subset of a topological space
• Limit of a sequence – Value to which tends an infinite sequence
• Limit point of a set – Cluster point in a topological spacePages displaying short descriptions of redirect targets
• Subsequential limit – The limit of some subsequence

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• This article incorporates material from Adherent point on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.