For every topological T1-space 
, the following conditions are equivalent.
1. 
is regular and second countable,
2. 
is separable and metrizable.
3. 
is homeomorphic to a subspace of the Hilbert cube.
Urysohn's Metrization Theorem
This entry contributed by Margherita Barile
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References
Willard, S. General Topology. Reading, MA: Addison-Wesley, p. 166, 1970.Referenced on Wolfram|Alpha
Urysohn's Metrization TheoremCite this as:
Barile, Margherita. "Urysohn's Metrization Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/UrysohnsMetrizationTheorem.html