Definition:Connected (Topology)

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

Definition

Topological Space

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

$T$ is connected if and only if there exists no continuous surjection from $T$ onto a discrete two-point space.

Set of Topological Space

$H$ is a connected set of $T$ if and only if it is not the union of any two non-empty separated sets of $T$.

Points in Topological Space

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

Let $a, b \in S$.

Then $a$ and $b$ are connected (in $T$) if and only if there exists a connected set in $T$ containing both $a$ and $b$.

Also see

• Equivalence of Definitions of Connected Topological Space

• Definition:Component (Topology)

• Definition:Disconnected (Topology)

• Results about connected spaces can be found here.