T0 Space is not necessarily T1 Space/Proof 1

Theorem

Let $T = \struct {S, \tau}$ be a be a $T_0$ space.

Then it is not necessarily the case that $T$ is a $T_1$ space.

Proof

Proof by Counterexample:

Let $T = \struct {S, \tau_p}$ be a particular point space such that $S$ is not a singleton.

From Particular Point Space is $T_0$, we have that $T$ is a $T_0$ space.

From Non-Trivial Particular Point Topology is not $T_1$, $T$ is not a $T_1$ space.

The result follows.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $2$: Separation Axioms: Completely Regular Spaces