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..