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

[math]\displaystyle{ x \in \operatorname{Cl}_X A }[/math] if and only if for all open subsets [math]\displaystyle{ U \subseteq X, }[/math] if [math]\displaystyle{ x \in U \text{ then } U \cap A \neq \varnothing. }[/math]

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 [math]\displaystyle{ x }[/math] contains at least one point of [math]\displaystyle{ A }[/math] different from [math]\displaystyle{ x. }[/math] Thus every limit point is an adherent point, but the converse is not true. An adherent point of [math]\displaystyle{ A }[/math] is either a limit point of [math]\displaystyle{ A }[/math] or an element of [math]\displaystyle{ A }[/math] (or both). An adherent point which is not a limit point is an isolated point.

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

Examples and sufficient conditions

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

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

Adherent points and subspaces

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

Proof

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

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

Adherent points and sequences

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

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..