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.

Examples and sufficient conditions

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 ${\displaystyle M.}$

Adherent points and subspaces

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 ${\displaystyle Y.}$

Proof

By assumption, ${\displaystyle S\subseteq X\subseteq Y}$ and ${\displaystyle x\in X.}$ Assuming that ${\displaystyle x\in \operatorname {Cl} _{X}S,}$ let ${\displaystyle V}$ be a neighborhood of ${\displaystyle x}$ in ${\displaystyle Y}$ so that ${\displaystyle x\in \operatorname {Cl} _{Y}S}$ will follow once it is shown that ${\displaystyle V\cap S\neq \varnothing .}$ The set ${\displaystyle U:=V\cap X}$ is a neighborhood of ${\displaystyle x}$ in ${\displaystyle X}$ (by definition of the subspace topology) so that ${\displaystyle x\in \operatorname {Cl} _{X}S}$ implies that ${\displaystyle \varnothing \neq U\cap S.}$ Thus ${\displaystyle \varnothing \neq U\cap S=(V\cap X)\cap S\subseteq V\cap S,}$ as desired. For the converse, assume that ${\displaystyle x\in \operatorname {Cl} _{Y}S}$ and let ${\displaystyle U}$ be a neighborhood of ${\displaystyle x}$ in ${\displaystyle X}$ so that ${\displaystyle x\in \operatorname {Cl} _{X}S}$ will follow once it is shown that ${\displaystyle U\cap S\neq \varnothing .}$ By definition of the subspace topology, there exists a neighborhood ${\displaystyle V}$ of ${\displaystyle x}$ in ${\displaystyle Y}$ such that ${\displaystyle U=V\cap X.}$ Now ${\displaystyle x\in \operatorname {Cl} _{Y}S}$ implies that ${\displaystyle \varnothing \neq V\cap S.}$ From ${\displaystyle S\subseteq X}$ it follows that ${\displaystyle S=X\cap S}$ and so ${\displaystyle \varnothing \neq V\cap S=V\cap (X\cap S)=(V\cap X)\cap S=U\cap S,}$ as desired. ${\displaystyle \blacksquare }$

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 ${\displaystyle X.}$

Adherent points and sequences

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 ${\displaystyle S.}$

See also

• 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

Notes

Citations

1. Steen, p. 5; Lipschutz, p. 69; Adamson, p. 15.

References

• Adamson, Iain T., A General Topology Workbook, Birkhäuser Boston; 1st edition (November 29, 1995). ISBN 978-0-8176-3844-3.
• Apostol, Tom M., Mathematical Analysis, Addison Wesley Longman; second edition (1974). ISBN 0-201-00288-4
• Lipschutz, Seymour; Schaum's Outline of General Topology, McGraw-Hill; 1st edition (June 1, 1968). ISBN 0-07-037988-2.
• L.A. Steen, J.A.Seebach, Jr., Counterexamples in topology, (1970) Holt, Rinehart and Winston, Inc..
• This article incorporates material from Adherent point on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.