Function spaces on Corson-like compacta (2406.07452v2)

Abstract: For an index set $\Gamma$ and a cardinal number $\kappa$ the $\Sigma_{\kappa}$-product of real lines $\Sigma_{\kappa}(\mathbb{R}^{\Gamma})$ consist of all elements of $\mathbb{R}^{\Gamma}$ with $<\kappa$ nonzero coordinates. A compact space is $\kappa$-Corson if it can be embedded into $\Sigma_{\kappa}(\mathbb{R}^{\Gamma})$ for some $\Gamma$. We also consider a class of compact spaces wider than the class of $\omega$-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 $\kappa$-Corson compact spaces $K$ for regular, uncountable cardinal numbers $\kappa$ 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 $\omega$-Corson compact spaces $K$ are preserved by linear homeomorphisms of function spaces $C_{p}(K)$.