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 neighbourhood of x (or equivalently, every open neighborhood of x) contains at least one point of A. A point x ∈ X is an adherent point for A if and only if x is in the closure of A, thus
- if and only if for all open subsets if then
This definition differs from that of a limit point, in that for a limit point it is required that every neighborhood of 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 (xn)n∈N be such a sequence and let x be its limit. Then by definition of limit, for all neighbourhoods U of x there exists N ∈ N such that xn ∈ U for all n ≥ N. In particular, xN ∈ U and also xN ∈ 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.
See also
- Limit point – Cluster point in a topological space
- Closure (topology) – All points and limit points in a subset of a topological space
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.