Adherent point

In mathematics, an adherent point (also closure point or point of closure) is a slight generalization of the idea of a limit point.

If A is a subset of a topological space X, then a point x in X is an adherent point of A if 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.

This definition is more general than 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 fails. 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 subset of a metric space X which is bounded above, then sup S is adherent to S.
• For any subset S of a metric space M, S contains all of its adherent points if, and only if, S is closed in M.
• In the interval (a, b], a is an adherent point that is not in the interval.

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.