# Definition:Connected (Topology)/Set

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## Definition

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

Let $H \subseteq S$ be a non-empty subset of $S$.

### Definition 1

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

### Definition 2

$H$ is a **connected set of $T$** if and only if it is not **disconnected in $T$**.

### Definition 3

$H$ is a **connected set of $T$** if and only if:

- the topological subspace $\struct {H, \tau_H}$ of $T$ is a connected topological space.

## Also known as

A **connected set** of a topological space $T = \left({S, \tau}\right)$ is often found referred to as a **connected subset (of $T$)**.

Some sources refer to the concept of a **connected subspace**, which is no more than a **connected set** under the subspace topology.

On $\mathsf{Pr} \infty \mathsf{fWiki}$, if the distinction is required, it will be specified explicitly.

## Also see

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

## Sources

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- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**connected**(of a set)