Totally Separated Space is Totally Disconnected

Theorem

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

Then $T$ is totally disconnected.

Proof

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

Then for every $x, y \in S: x \ne y$ there exists a partition $U \mid V$ of $T$ such that $x \in U, y \in V$.

Suppose $T$ were not totally disconnected.

Then $\exists H \subseteq S: x, y \in U$ such that $H$ is connected.

But by definition of connected, there can be no partition $U \mid V$ of $T$ such that $x \in U, y \in V$.

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: Disconnectedness