Definition:Isolated Point (Topology)/Subset
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Definition
Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq S$ be a subset of $S$.
Definition 1
$x \in H$ is an isolated point of $H$ if and only if:
- $\exists U \in \tau: U \cap H = \set x$
That is, if and only if there exists an open set of $T$ containing no points of $H$ other than $x$.
Definition 2
$x \in H$ is an isolated point of $H$ if and only if $x$ is not a limit point of $H$.
That is, if and only if $x$ is not in the derived set of $H$.
Also see
- Results about isolated points can be found here.