# Definition:Connected (Topology)/Set

## Definition

Let $T = \struct {S, \tau}$ 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.

### Definition 4

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

- there do not exist disjoint, non-empty subsets $X$ and $Y$ of $H$ such that $X \cup Y = H$ such that:
- no limit point of $X$ is an element of $Y$
- no limit point of $Y$ is an element of $X$.

### Definition 5

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

- $H$ cannot be partitioned into $2$ non-empty subsets so that each subset has no element in common with the closure of the other.

## Also known as

A **connected set** of a topological space $T = \struct {S, \tau}$ 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**.