Locally Connected Space is not necessarily Connected

Theorem

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

Then it is not necessarily the case that $T$ is also a connected space.

Proof

Let $T$ be a discrete topological space with more than $1$ point.

From Discrete Space is Locally Connected, $T$ is a locally connected space.

From Non-Trivial Discrete Space is not Connected, $T$ is not a connected space.

Hence the result.

$\blacksquare$

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