Definition:Disconnected (Topology)/Set/Definition 2

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Definition

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

Let $H \subseteq S$ be a non-empty subset of $S$.


$H$ is a disconnected set of $T$ if and only if there exist open sets $U$ and $V$ of $T$ such that:

$H \subseteq U \cup V$
$H \cap U \cap V = \varnothing$
$U \cap H \ne \varnothing$

and:

$V \cap H \ne \varnothing$


Also see

  • Results about disconnected sets can be found here.