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Concept# Adherent point

Summary

In mathematics, an adherent point (also closure point or point of closure or contact point) 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 \in X is an adherent point for A if and only if x is in the closure of A, thus
:x \in \operatorname{Cl}_X A if and only if for all open subsets U \subseteq X, if x \in U \text{ then } U \cap A \neq \varnothing.
This definition differs from that of a limit point of a set, in that for a limit point it is required that every neighborhood of x contains at least one point of A x. Thus every limit point is an adherent point, but the converse is not true. An adherent point of A is either

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