Definition:Adherent Point

Definition

Let $T = \left({X, \tau}\right)$ be a topological space.

Let $A \subseteq X$.

Definition by Open Neighborhood

A point $x \in X$ is called an adherent point of $A$ if every open neighborhood $U$ of $x$ satisfies $A \cap U \ne \varnothing$.

Definition from Closure

Equivalently, $x$ is an adherent point of $A$ if $x$ belongs to the closure of $A$.

Also see

• Equivalence of Adherent Point Definitions

• Condensation Point
• $\omega$-Accumulation Point
• Limit Point

• Relationship between Limit Point Types

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$: Limit Points