A new characterization of spaces with locally countable \(sn\)-networks (Q6486675)

The author characterizes spaces with locally countable \(sn\)-networks or weak bases. The main result states that a space \(X\) has a locally countable \(sn\)-network if and only if \(X\) is a compact-covering (pseudo-sequence-covering or subsequence-covering) compact and \(ss\)-image of a metric space, which is equivalent to \(X\) be a sequentially quotient \(\pi\)- and \(ss\)-image of a metric space. An analogous statement holds for spaces with locally countable weak bases.