Function spaces on Corson-like compacta
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 $C_{p}(K)$.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2024
- DOI:
- 10.48550/arXiv.2406.07452
- arXiv:
- arXiv:2406.07452
- Bibcode:
- 2024arXiv240607452Z
- Keywords:
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- Mathematics - General Topology