Definition:Connected (Topology)

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This page is about connectedness in topology. For other uses, see Definition:Connected.

Definition

Topological Space

Let $T = \left({S, \tau}\right)$ 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 $T = \left({S, \tau}\right)$ 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

  • Results about connected spaces can be found here.