On the theory of $K$-analytic spaces

Main results are the following. Let $X$ be a regular $K$-analytic space. Then (1) $X$ is hereditarily Lindelöf and hereditarily separable if and only if there does not exist any strongly increasing transfinite sequence $\{f_{\alpha}\colon\alpha<\omega_1\}$ of functions of the first Baire class; (2) every directed acontinuous covering of $X$ by $G_\delta$ sets lias a countable subcovering.