Adherent point

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 (i.e. every open set containing x) contains at least one point of A. A point x is an adherent point for A if and only if x is in the closure of A, thus

$x\in \operatorname {Cl} ({A})\iff \forall U\subset X{\text{ open}}:x\in U\implies U\cap A\neq \emptyset .$

This definition differs from that of a limit point, in that for a limit point it is required that every open set containing 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

• If S is a non-empty subset of R which is bounded above, then sup S is adherent to S.
• A subset S of a metric space M contains all of its adherent points if, and only if, S is (sequentially) closed in M.
• In the interval (a, b], a is an adherent point that is not in the interval, with usual topology of R.
• 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 (x_n)_n∈N be such a sequence and let x be its limit. Then by definition, for all open neighbourhoods U of x there exists N ∈ N such that x_n ∈ U for all n ≥ N. In particular, x_N ∈ U and also x_N ∈ 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 R. Then the only sequence in S is the constant sequence (0) whose limit is 0, but 0 is not a limit point of S; it is only an adherent point of S.

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.