Compactly metrizable spaces and a theorem on generalized strong Σ -spaces

We prove that any jointly metrizable on compacta space X with a countable kω-network has a countable network (Corollary 2.3). Theorem 3.3 states that, for any continuous mapping g:X→Y of a paracompact p-space X onto a jointly metrizable on compacta space Y, there exist a metrizable space Z, a perfect mapping f:X→Z, and a continuous mapping h:Z→Y such that g=h∘f. It follows that a Lindelöf Σ-space is a JCM-space if and only if it has a countable network.