# Definition:Connected (Topology)

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*This page is about Connected in the context of topology. For other uses, see 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 $T = \struct {S, \tau}$ 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

- Results about
**connected spaces**can be found here.