Hilbert cube

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The Hilbert cube is homeomorphic to countably many copies of the unit interval $[0,1]$, sometimes written as $H=[0,1]^{\infty}$.

Properties

Proposition: The Hilbert cube is metrizable; define a metric for $\vec{x}=(x_1,x_2,\ldots)$ and $\vec{y}=(y_1,y_2,\ldots)$ in the Hilbert cube by $$d_{\infty}(\vec{x},\vec{y})=\displaystyle\sum_{k=1}^{\infty} \dfrac{|x_i-y_1|}{2^k}.$$

Proof: We must prove that $d$ is a metric. █

Characterization of when 2^X is homogeneous
Characterization of when C(X) is homogeneous