Adherent point

Short description: Point that belongs to the closure of some given subset of a topological space

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

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

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 $x$ 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 $M.$

Adherent points and subspaces

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

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

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 $X.$

Adherent points and sequences

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 ${\left(x_n\right)}_{n\in \mathbb{\mathbb{N}}}$ be such a sequence and let $x$ be its limit. Then by definition of limit, for all neighbourhoods $U$ of $x$ there exists $n\in \mathbb{\mathbb{N}}$ such that $x_n\in U$ for all $n\ge N.$ 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 $0,0,\dots$ whose limit is $0,$ but $0$ is not a limit point of $S;$ it is only an adherent point of $S.$

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 – Point in a topological space
• Subsequential limit – The limit of some subsequence

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..