Definition:Disconnected (Topology)/Set

Definition

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

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

Definition 1

$H$ is a disconnected set of $T$ if and only if it is not a connected set of $T$.

Definition 2

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

$H \subseteq U \cup V$

$H \cap U \cap V = \O$

$U \cap H \ne \O$

$V \cap H \ne \O$

Definition 3

$H$ is a disconnected set of $T$ if and only if there exist non-empty subsets $U$ and $V$ of $H$ such that all of the following hold:

$H = U \cup V$

no limit point of $U$ is an element of $V$

no limit point of $V$ is an element of $U$.

Also see

• Equivalence of Definitions of Disconnected Set

• Results about disconnected sets can be found here.