Existence of Non-Locally Connected Space where Components and Quasicomponents are Equal

Theorem

There exists at least one example of a topological space which is not locally connected, but whose components and quasicomponents are equal.

Proof

Let $T$ be the Arens-Fort space.

From Arens-Fort Space is not Locally Connected, $T$ is not a locally connected space.

The result follows from Components and Quasicomponents of Arens-Fort Space are Equal.

$\blacksquare$

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