Definition:Adherent Point
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Definition
Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq S$.
Definition by 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 by 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 by 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
- Results about adherent points can be found here.