Totally Separated Space is Totally Disconnected

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Theorem

Let $T = \left({S, \tau}\right)$ be a topological space which is totally separated.

Then $T$ is totally disconnected.


Proof

Let $T = \left({S, \tau}\right)$ 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.

$\blacksquare$


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