Totally Disconnected and Locally Connected Space is Discrete

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Theorem

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

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


Proof

So, let $T = \left({S, \tau}\right)$ 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 $\mathcal B$ of $T$ such that every element of $\mathcal B$ is a component of $T$.

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

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 $S$.

$\blacksquare$


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