Hilbert Cube is Completely Normal

Theorem

Let $M = \struct {I^\omega, d_2}$ be the Hilbert cube.

Then $M$ is a completely normal space.

Proof

We have that Hilbert Cube is Metric Space.

The result follows from Metric Space is Completely Normal.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $38$. Hilbert Cube: $2$