Another question on second countable spaces

Question

Let $X$ have countable chain condition and point countable base. Is $X$ second countable? I thing it don't need to be. However I have no examples at hand.

Thanks for your help.

Answer 1

Your suspicion is correct, though the absolute examples that I have are a little complicated. Ronnie Levy and Mikhail Matveev, ‘Spaces with $\sigma$-$n$-linked topologies as special subspaces of separable spaces’, Commentationes Mathematicae Universitatis Carolinae, Vol. $40$ ($1999$), No. $3$, $561$-$570$, Example $1.8$ cites some examples from Juris Steprāns and Stephen Watson, ‘Cellularity of first countable spaces’, Topology and its Applications, $28 (1988)$, $141$-$145$, of ccc, non-separable subspaces of the Pixley-Roy space built from ${}^\omega\omega$, the irrationals; since they are not separable, they cannot be second countable. They are metacompact Moore spaces, so they have point-countable bases.