Totally Disconnected and Locally Connected Space is Discrete

Theorem

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

Then $T$ is the discrete space on $S$.

So, let $T = \struct {S, \tau}$ be that topological space which is both totally disconnected and locally connected.

As $T$ is totally disconnected, every point is a component and therefore closed.

As $T$ is locally connected, there exists a basis $\BB$ of $T$ such that every element of $\BB$ is a component of $T$.

In order for $T$ to be covered by $\BB$, every singleton subset of $T$ must be in $\BB$.

From the definition of topology, the union of any number of these singleton sets is an open set of $T$.

That is, every subset of $S$ is open in $T$.

That is, every element of the power set of $S$ is open in $T$.

This is precisely the definition of the discrete space on $S$.

$\blacksquare$

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