Urysohn's Metrization Theorem

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

Then $T$ is metrizable.


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

It follows that $T$ is metrizable by definition.


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.