Definitions

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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