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Application of selection principles in the study of the properties of function spaces

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Abstract

For a Tychonoff space X, we denote by C p (X) the space of all real-valued continuous functions on X with the topology of pointwise convergence.

In this paper we prove that:

  • If every finite power of X is Lindelöf then C p (X) is strongly sequentially separable iff X is \({\gamma}\)-set.

  • \({B_{\alpha}(X)}\) (= functions of Baire class \({\alpha}\) (\({1 < \alpha \leq \omega_1}\)) on a Tychonoff space X with the pointwise topology) is sequentially separable iff there exists a Baire isomorphism class \({\alpha}\) from a space X onto a \({\sigma}\)-set.

  • \({B_{\alpha}(X)}\) is strongly sequentially separable iff \({iw(X)=\aleph_0}\) and X is a \({Z^{\alpha}}\)-cover \({\gamma}\)-set for \({0 < \alpha \leq \omega_1}\).

  • There is a consistent example of a set of reals X such that C p (X) is strongly sequentially separable but B1(X) is not strongly sequentially separable.

  • B(X) is sequentially separable but is not strongly sequentially separable for a \({\mathfrak{b}}\)-Sierpiński set X.

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  1. Krasovskii Institute of Mathematics and Mechanics, Ural Federal University, Ural State University of Economics, Yekaterinburg, Russia

    A. V. Osipov

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Osipov, A.V. Application of selection principles in the study of the properties of function spaces. Acta Math. Hungar. 154, 362–377 (2018). https://doi.org/10.1007/s10474-018-0800-4

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  • DOI: https://doi.org/10.1007/s10474-018-0800-4

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