Digging into the classes of $\kappa$-Corson compact spaces

Abstract

For any cardinal number $\kappa$ and an index set $\Gamma$, $\Sigma_\kappa$-product of real lines consists of elements of ${\mathbb R}^\Gamma$ having $<\kappa$ nonzero coordinates. A compact space $K$ is $\kappa$-Corson compact if it can be embedded into such a space for some $\Gamma$. The class of ($\omega_1$-)Corson compact spaces has been intensively studied over last decades. We discuss properties of $\kappa$-Corson compacta for various cardinal numbers $\kappa$ as well as properties of related Boolean algebras and spaces of continuous functions. We present here a detailed discussion of the class of $\omega$-Corson compacta extending the results of Nakhmanson and Yakovlev. For $\kappa>\omega$, our results on $\kappa$-Corson compact spaces are related to the line of research originated by Kalenda and Bell and Marciszewski, and continued by Bonnet, Kubis and Todorcevic in their recent paper.