Is a topological space pseudo-metrizable if and only if it is regular and paracompact ?
I have a link to a reference , "General Topology" , Kelley , page 127 , theorem 18 . Also note a paracompact space has a locally finite base, and so as required there, a sigma-locally finite base.
On page 128, the proof of lemma 20 shows, if the assumption that the space is T1 is dropped, that a regular paracompact space is pseudo-metrizable.
No, paracompact is not enough. Consider the Sorgenfrey Line or the Double arrow space.
We really need $\sigma$-locally finite base or a $\sigma$-discrete base.