Urysohn's Metrization Theorem

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Theorem

Let $T = \struct {S, \tau}$ be a topological space which is regular and second-countable.


Then $T$ is metrizable.


Proof

From Regular Second-Countable Space is Homeomorphic to Subspace of Hilbert Cube:

$T$ is homeomorphic to a subspace of the Hilbert cube $\struct{I^\omega, d_2}$

where $d_2$ is the metric: $\ds \forall x = \sequence {x_i}, y = \sequence {y_i} \in I^\omega: \map {d_2} {x, y} := \paren {\sum_{k \mathop \in \N_{>0} } \paren {x_k - y_k}^2}^{\frac 1 2}$


By definition of metrizable topology:

$\struct{I^\omega, \tau_{d_2}}$ is metrizable

where $\tau_{d_2}$ is the topology induced by $d_2$.


From Subspace of Metrizable Space is Metrizable Space:

$T$ is homeomorphic to a metrizable space

From Topological Space Homeomorphic to Metrizable Space is Metrizable Space:

$T$ is metrizable

$\blacksquare$


Also see


Source of Name

This entry was named for Pavel Samuilovich Urysohn.


Historical Note

This form of Urysohn's Metrization Theorem was actually proved by Andrey Nikolayevich Tychonoff in $1926$.

What Urysohn had shown, in a posthumous $1925$ paper, was that every second-countable normal Hausdorff space is metrizable.


Sources

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