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

## TL;DR

This paper explores the properties of continuous images of ch-complete spaces, establishing conditions under which compact subsets are scattered and analyzing the cardinality constraints of their images.

## Contribution

It provides new equivalences relating scattered compact sets, continuous images, and cardinality bounds in ch-complete and K-analytic spaces.

## Key findings

- Equivalence of conditions for scattered compact sets in continuous images.
- Cardinality bounds for images of certain spaces under continuous maps.
- Characterization of countability in K-analytic spaces with a unique non-isolated point.

## 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.