T4 and first countable topology that is non metrizable

Question

Does anyone know any example of such topology?

Answer 1

The first uncountable ordinal with the order topology is normal, first countable and even locally metrizable - but it is not metrizable, since it is non-compact despite the fact that it is sequentially compact

Answer 2

The Sorgenfrey Line is $T_4$ and first-countable. It cannot be metrizable since it has a countable dense subset $\Bbb Q$ but has no countable base, that means it is separable without being second-countable.