Scattered compact sets in continuous images of \v{C}ech-complete spaces

Abstract

Assume hat a functionally Hausdorff space $X$ is a continuous image of a \v{C}ech complete space $P$ with Lindel\"of number $l(P)<\mathfrak c$. Then the following conditions are equivalent: (i) every compact subset of $X$ is scattered, (ii) for every continuous map $f:X\to Y$ to a functionally Hausdorff space $Y$ the image $f(X)$ has cardinality $|f(X)|\le \max\{l(P),\psi(Y)\}$, (iii) no continuous map $f:X\to[0,1]$ is surjective. Also we prove the equivalence of the conditions: (a) $\omega_1<\mathfrak b$, (b) a K-analytic space $X$ (with a unique non-isolated point) is countable if and only if every compact subset of $X$ is countable.