Definitions

adherent

Adherent point

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

Let X be a topological space and Asubset X be a subset. A point xin X is an adherent point for 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 which is not a limit point is an isolated point.

References

  • L.A. Steen, J.A.Seebach, Jr., Counterexamples in topology, (1970) Holt, Rinehart and Winston, Inc..

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