Definition:Connected (Topology)/Set/Definition 1

Definition

Let $T = \struct {S, \tau}$ be a topological space.

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

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

Also see

• Equivalence of Definitions of Connected Set

• Results about connected sets can be found here.

Sources

• 1953: Walter Rudin: Principles of Mathematical Analysis ... (previous) ... (next): $2.45$
• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Closures and Interiors
• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness