My definition for $T_3$ space is a topological space such that 'For any point $x$ and a closed subset $A$, there exist nonoverlapping neighborhoods'.
(This is the definition of a regular space in "Topology - Munkres")
(p.215) Urysohn Metrization theorem: Every second countable $T_3$ space $X$ is metrizable.
Is it true? I can prove this with an additional condition, that is, $X$ is $T_0$. However, i cannot prove this without this condition. How do i prove this without this condition?
What are necessary conditions for the Urysohn Metrization theorem?