Hilbert Cube is Second-Countable
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Theorem
Let $M = \struct {I^\omega, d_2}$ be the Hilbert cube.
Then $M$ is a second-countable space.
Proof
From Hilbert Cube is Separable, $M$ is a separable space.
We also have that Hilbert Cube is Metric Space.
The result follows from Separable Metric Space is Second-Countable.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $38$. Hilbert Cube: $3$