Metric Space is Completely Normal
Jump to navigation
Jump to search
Theorem
Let $M = \struct {A, d}$ be a metric space.
Then $M$ is a completely normal space.
Proof
By definition, a topological space is completely normal space if and only if it is:
We have that:
- a Metric Space is $T_5$
- a Metric Space is $T_2$ (Hausdorff)
- a $T_2$ (Hausdorff) Space is a $T_1$ (Fréchet) Space.
Hence the result.
$\blacksquare$
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