adherent point

In mathematics, an adherent point (also closure point or point of closure or contact point) Steen, p. 5; Lipschutz, p. 69; Adamson, p. 15. 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
:$x \in \operatorname_X A$ if and only if for all open subsets $U \subseteq X,$ if $x \in U \text U \cap A \neq \varnothing.$

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 $x$ contains at least one point of $A$ $x.$ 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 $\R$ 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 Topological space, topology of $\R.$

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 $Y.$

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

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 $\left(x_n\right)_$ be such a sequence and let $x$ be its limit. Then by definition of limit, for all neighbourhoods $U$ of $x$ there exists $n \in \N$ 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 = \$ as a subset of $\R.$ 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 $S.$

See also

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Notes

Citations

References

* Adamson, Iain T.,
A General Topology Workbook
', Birkhäuser Boston; 1st edition (November 29, 1995). .
* Apostol, Tom M., ''Mathematical Analysis'', Addison Wesley Longman; second edition (1974).
* Lipschutz, Seymour; ''Schaum's Outline of General Topology'', McGraw-Hill; 1st edition (June 1, 1968). .
* L.A. Steen, J.A.Seebach, Jr., ''Counterexamples in topology'', (1970) Holt, Rinehart and Winston, Inc..
*{{PlanetMath attribution, urlname=adherentpoint, title=Adherent point

General topology