Definition:Connected (Topology)/Set/Definition 3

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:

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

Also see

• Equivalence of Definitions of Connected Set

• Results about connected sets can be found here.

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $6.2$: Connectedness: Corollary $6.2.4$
• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness