Definition:Adherent Point

Definition

Let $T = \struct {S, \tau}$ be a topological space.

Let $H \subseteq S$.

Definition from Neighborhood

A point $x \in S$ is an adherent point of $H$ if and only if every neighborhood $N$ of $x$ satisfies:

$H \cap N \ne \O$

Definition from Open Neighborhood

A point $x \in S$ is an adherent point of $H$ if and only if every open neighborhood $U$ of $x$ satisfies:

$H \cap U \ne \O$

Definition from Closure

A point $x \in S$ is an adherent point of $H$ if and only if $x$ is an element of the closure of $H$.

Also see

• Equivalence of Definitions of Adherent Point
• Definition:Condensation Point
• Definition:Omega-Accumulation Point
• Definition:Limit Point of Set

• Relationship between Limit Point Types

• Results about adherent points can be found here.