Scattered compact sets in continuous images of Čech-complete spaces

Abstract

Assume that a functionally Hausdorff space X is a continuous image of a Čech complete space P with Lindelöf number l(P)<c. Then the following conditions are equivalent: (i) every compact subset of X is scattered, (ii) for every continuous map f:X→Y to a functionally Hausdorff space Y the image f(X) has cardinality not exceeding max⁡{l(P),ψ(Y)}, (iii) no continuous map f:X→[0,1] is surjective. We also prove the equivalence of the conditions: (a) ω1<b, and (b) a K-analytic space X (with a unique non-isolated point) is countable iff every compact subset of X is countable.