Definition:Disconnected (Topology)

This page is about Disconnected in the context of topology. For other uses, see Disconnected.

Definition

Topological Space

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

Definition $1$

$T$ is disconnected if and only if $T$ is not connected.

Definition $2$

$T$ is disconnected if and only if there exist non-empty open sets $U, V \in \tau$ such that:

$S = U \cup V$

$U \cap V = \O$

That is, if there exists a partition of $S$ into open sets of $T$.

Subset of Topological Space

Let $T = \struct {S, \tau}$ 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 it is not a connected set of $T$.

Points in Topological Space

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

Let $a, b \in S$.

Then $a$ and $b$ are disconnected (in $T$) if and only if they are not connected (in $T$).

Also see

• Results about disconnected spaces can be found here.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): disconnected