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

• Equivalence of Definitions of Connected Set

• Results about connected sets can be found here.