# Adherent point

In mathematics, an **adherent point** (also **closure point** or **point of closure** or **contact point**)[1] of a subset *A* of a topological space *X*, is a point *x* in *X* such that 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 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*.

## Notes

- Steen, p. 5; Lipschutz, p. 69; Adamson, p. 15.

## 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.*