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