Definition:Disconnected (Topology)
Jump to navigation
Jump to search
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