Are Eberlein-Grothendieck scattered spaces $σ$-discrete?

Details Are Eberlein-Grothendieck scattered spaces $σ$-discrete? Are Eberlein-Grothendieck scattered spaces $σ$-discrete?

2012-02-28

arxiv.org/abs/1202.6310v1

Mathematics

General Topology

Scientific paper

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.

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