Definition:Connected (Topology)/Topological Space/Definition 1

Definition

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

$T$ is connected if and only if it admits no separation.

That is, $T$ is connected if and only if there exist no open sets $A, B \in \tau$ such that $A, B \ne \O$, $A \cup B = S$ and $A \cap B = \O$.

Also see

• Equivalence of Definitions of Connected Topological Space

• Results about connected spaces can be found here.

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $6.2$: Connectedness: Proposition $6.2.3$
• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness