A compact space K is Corson compact if and only if Cp(K) has a dense lc-scattered subspace

Abstract

We establish that a compact space K is Corson compact if and only if it can be embedded in a topological group which is a W-space in the sense of Gruenhage. As a application of this result we show that K is Corson compact if and only if it embeds in Cp(X) for a Lindelöf scattered space X. We also prove that a zero-dimensional compact space K is Corson if and only if Cp(K,{0,1}) is a continuous image of a Lindelöf scattered space. Besides, a zero-dimensional countably compact space X is ω-monolithic if and only if Cp(X) has a dense functionally countable subspace.