# Equivalence of Definitions of Totally Separated Space

## Theorem

The following definitions of the concept of **Totally Separated Space** are equivalent:

### Definition 1

A topological space $T = \struct {S, \tau}$ is **totally separated** if and only if:

- For every $x, y \in S: x \ne y$ there exists a separation $U \mid V$ of $T$ such that $x \in U, y \in V$.

### Definition 2

A topological space $T = \struct {S, \tau}$ is **totally separated** if and only if each of its quasicomponents is a singleton set.

## Proof

Let $T = \struct {S, \tau}$ be **totally separated** by definition 1:

- For every $x, y \in S: x \ne y$ there exists a separation $U \mid V$ of $T$ such that $x \in U, y \in V$.

Aiming for a contradiction, suppose there exist $x, y \in S$ such that $x \ne y$ such that $x$ and $y$ are in the same quasicomponent of $T$.

Then, by definition of quasicomponent, every separation of $T$ includes a single open set $U \in \tau$ which contains both $x$ and $y$.

But this contradicts our stipulation that there exists a separation $U \mid V$ of $T$ such that $x \in U, y \in V$.

So there exists no quasicomponent of $T$ with more than one element.

That is, $T = \struct {S, \tau}$ is **totally separated** by definition 2.

$\Box$

Let $T = \struct {S, \tau}$ be **totally separated** by definition 2:

- Each of its quasicomponents is a singleton set.

Aiming for a contradiction, suppose there exist $x, y$ such that there exists no separation $U \mid V$ of $T$ such that $x \in U, y \in V$.

That is, each separation of $T$ contains an open set $U \in \tau$ which contains both $x$ and $y$.

That is, there exists a quasicomponent which contains both $x$ and $y$.

But this contradicts our stipulation that every quasicomponent of $T$ is a singleton.

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

That is, $T = \struct {S, \tau}$ is **totally separated** by definition 1.

$\blacksquare$

## 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