Function spaces on Corson-like compacta

2024/06/11 by Zakrzewski, Krzysztof
#FOS: Mathematics #General Topology (math.GN)

paper · doi:10.48550/arxiv.2406.07452

Abstract

For an index set Γ and a cardinal number κ the Σ_κ-product of real lines Σ_κ(ℝ^Γ) consist of all elements of ℝ^Γ with <κ nonzero coordinates. A compact space is κ-Corson if it can be embedded into Σ_κ(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).

Related

• Digging into the classes of κ-Corson compact spaces
• On the class of NY compact spaces of finitely supported elements and related classes
• Function spaces on separable compact lines
• A note on condensations of function spaces onto σ-compact and analytic spaces
• κ-spaces