Quasicomponents and Components are Equal in Locally Connected Space

Theorem

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

Then $A \subseteq S$ is a component of $T$ if and only if $A \subseteq S$ is a quasicomponent of $T$.

Proof

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Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness