Function spaces on Corson-like compacta

Abstract

For an index set $Γ$ and a cardinal number $κ$ the $Σ_κ$-product of real lines $Σ_κ(\mathbb{R}^Γ)$ consist of all elements of $\mathbb{R}^Γ$ with $<κ$ nonzero coordinates. A compact space is $κ$-Corson if it can be embedded into $Σ_κ(\mathbb{R}^Γ)$ for some $Γ$. We also consider a class of compact spaces wider than the class of $ω$-Corson compact spaces, investigated by Nakhmanson and Yakovlev as well as Marciszewski, Plebanek and Zakrzewski called $NY$ compact spaces. For a Tychonoff space $X$, let $C_{p}(X)$ be the space of real continuous functions on the space $X$, endowed with the pointwise convergence topology. We present here a characterisation of $κ$-Corson compact spaces $K$ for regular, uncountable cardinal numbers $κ$ in terms of function spaces $C_{p}(K)$, extending a theorem of Bell and Marciszewski and a theorem of Pol. We also prove that classes of $NY$ compact spaces and $ω$-Corson compact spaces $K$ are preserved by linear homeomorphisms of function spaces $C_{p}(K)$.