Are Eberlein-Grothendieck scattered spaces $\sigma$-discrete?

TL;DR

This paper investigates whether Eberlein-Grothendieck scattered spaces are $\sigma$-discrete, establishing conditions under which they are $\sigma$-discrete or $\sigma$-compact, especially for spaces with certain height and compactness properties.

Contribution

It proves that under specific conditions, Eberlein-Grothendieck scattered spaces are $\sigma$-discrete or $\sigma$-compact, advancing understanding of their structure.

Findings

Spaces with $w(K) extless\omega_1$ and height $ extless\omega_1$ are $\sigma$-discrete.

Every Lindelöf Čech-complete scattered space is $\sigma$-compact.

Conditions involving local compactness or countability ensure $\sigma$-discreteness.

Abstract

A space $X$ is Eberlein-Grothendieck if $X\subset C_p(K)$ for some compact space $K.$ In this paper we address the problem of whether such a space $X$ is $\sigma$-discrete whenever it is scattered. We show that if $w(K)\leq\omega_1$ then $X$ is $\sigma$-dicrete whenever $X$ has height $\omega_1$ and it is locally compact or locally countable. It is also proved that every Lindel\"of \v{C}ech-complete scattered space is $\sigma$-compact.