Totally analytic spaces under $V=L$

Abstract:

The following results obtain under the axiom of constructibility $(V = L)$: Assume that every subset of a topological space $X$ is analytic. Then $X$ is $\sigma$-left-separatcd. Moreover, if the character of $X$ is $\leqslant {\omega _1}$, then $X$ is $\sigma$-discrete. Assume that $X$ is a perfectly normal space of character $\leqslant {\omega _1}$ such that every subset of $X$ belongs to the $\sigma$-algebra generated by the analytic subsets of $X$. Then $X$ is $\sigma$-discrete.