“Adherent point”的意思、由来-开放百科全书

词条

Adherent point

释义

Examples
Notes
References

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 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 closed in M.
In the interval (a, b], a is an adherent point that is not in the interval, with usual topology of R.

Notes

1. ^ 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..
{{PlanetMath attribution|urlname=adherentpoint|title=Adherent point}}

1 : General topology

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。