Discrete Space is Locally Connected

Theorem

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

Then $T$ is locally connected.

Proof

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

From Discrete Space is Locally Path-Connected, $T$ is locally path-connected.

From Locally Path-Connected Space is Locally Connected, it follows that $T$ is locally connected.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $1 \text { - } 3$. Discrete Topology: $10$