Definition:Isolated Point (Topology)

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Definition

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


Isolated Point of Subset

$x \in H$ is an isolated point of $H$ if and only if:

$\exists U \in \tau: U \cap H = \left\{{x}\right\}$

That is, if and only if there exists an open set of $T$ containing no points of $H$ other than $x$.


Isolated Point of Space

When $H = S$ the definition applies to the entire topological space $T = \left({S, \tau}\right)$:


$x \in S$ is an isolated point of $T$ if and only if:

$\exists U \in \tau: U = \left\{{x}\right\}$

That is, if and only if there exists an open set of $T$ containing no points of $S$ other than $x$.


The set of isolated points of $H \subseteq S$ can be denoted $H^i$.


Also see

  • Results about isolated points can be found here.