Metric Space fulfils all Separation Axioms

Theorem

Let $M = \struct {A, d}$ be a metric space.

Then $M$, considered as a topological space, fulfils all separation axioms:

$M$ is a $T_0$ (Kolmogorov) space

$M$ is a $T_1$ (Fréchet) space

$M$ is a $T_2$ (Hausdorff) space

$M$ is a semiregular space

$M$ is a $T_{2 \frac 1 2}$ (completely Hausdorff) space

$M$ is a $T_3$ space

$M$ is a regular space

$M$ is an Urysohn space

$M$ is a $T_{3 \frac 1 2}$ space

$M$ is a Tychonoff (completely regular) space

$M$ is a $T_4$ space

$M$ is a normal space

$M$ is a $T_5$ space

$M$ is a completely normal space

$M$ is a perfectly $T_4$ space

$M$ is a perfectly normal space

Proof

We have that:

A metric space is a $T_2$ (Hausdorff) space.

A metric space is a perfectly $T_4$ space.

A metric space is a $T_5$ space.

‎ From Sequence of Implications of Separation Axioms we then have:

$T_5$ space is $T_4$ space.

$T_2$ (Hausdorff) Space is $T_1$ (Fréchet) Space.

$T_1$ (Fréchet) Space is $T_0$ (Kolmogorov) Space.

By definition, a perfectly normal space is:

a perfectly $T_4$ space

a $T_1$ (Fréchet) space

So $M$ is a perfectly normal space.

By definition, a completely normal space is:

a $T_5$ space

a $T_1$ (Fréchet) space

So $M$ is a completely normal space.

Then from Sequence of Implications of Separation Axioms we can complete the chain:

Completely Normal Space is Normal Space.

Normal Space is Tychonoff (Completely Regular) Space.

Completely Regular (Tychonoff) implies $T_{3 \frac 1 2}$ by definition.

Completely Regular (Tychonoff) Space is Regular Space.

Completely Regular (Tychonoff) Space is Urysohn Space.

Regular implies $T_3$ by definition.

Regular Space is Completely Hausdorff Space.

Regular Space is Semiregular Space.

$\blacksquare$

There are other chains of implications which can be used, but the above are sufficient to prove the hypothesis.

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces