Definition:Connected (Topology)/Set

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

• Equivalence of Definitions of Connected Set

• 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)