Basic properties of $X$ for which the space $C_p(X)$ is distinguished
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- by Jerzy Ka̧kol and Arkady Leiderman HTML | PDF
- Proc. Amer. Math. Soc. Ser. B 8 (2021), 267-280
Abstract:
In our paper [Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 86–99] we showed that a Tychonoff space $X$ is a $\Delta$-space (in the sense of R. W. Knight [Trans. Amer. Math. Soc. 339 (1993), pp. 45–60], G. M. Reed [Fund. Math. 110 (1980), pp. 145–152]) if and only if the locally convex space $C_{p}(X)$ is distinguished. Continuing this research, we investigate whether the class $\Delta$ of $\Delta$-spaces is invariant under the basic topological operations.
We prove that if $X \in \Delta$ and $\varphi :X \to Y$ is a continuous surjection such that $\varphi (F)$ is an $F_{\sigma }$-set in $Y$ for every closed set $F \subset X$, then also $Y\in \Delta$. As a consequence, if $X$ is a countable union of closed subspaces $X_i$ such that each $X_i\in \Delta$, then also $X\in \Delta$. In particular, $\sigma$-product of any family of scattered Eberlein compact spaces is a $\Delta$-space and the product of a $\Delta$-space with a countable space is a $\Delta$-space. Our results give answers to several open problems posed by us [Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 86–99].
Let $T:C_p(X) \longrightarrow C_p(Y)$ be a continuous linear surjection. We observe that $T$ admits an extension to a linear continuous operator $\widehat {T}$ from $\mathbb {R}^X$ onto $\mathbb {R}^Y$ and deduce that $Y$ is a $\Delta$-space whenever $X$ is. Similarly, assuming that $X$ and $Y$ are metrizable spaces, we show that $Y$ is a $Q$-set whenever $X$ is.
Making use of obtained results, we provide a very short proof for the claim that every compact $\Delta$-space has countable tightness. As a consequence, under Proper Forcing Axiom every compact $\Delta$-space is sequential.
In the article we pose a dozen open questions.
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Additional Information
- Jerzy Ka̧kol
- Affiliation: Faculty of Mathematics and Informatics, A. Mickiewicz University, 61-614 Poznań, Poland; and Institute of Mathematics, Czech Academy of Sciences, Prague, Czech Republic
- MR Author ID: 96980
- ORCID: 0000-0002-8311-2117
- Email: kakol@amu.edu.pl
- Arkady Leiderman
- Affiliation: Department of Mathematics, Ben-Gurion University of the Negev, Beer Sheva, Israel
- MR Author ID: 214471
- ORCID: 0000-0002-2257-1635
- Email: arkady@math.bgu.ac.il
- Received by editor(s): March 23, 2021
- Received by editor(s) in revised form: June 15, 2021
- Published electronically: September 21, 2021
- Additional Notes: The research for the first author was supported by the GAČR project 20-22230L and RVO: 67985840. He was also supported by the Center for Advanced Studies in Mathematics of Ben-Gurion University of the Negev for financial support during his visit in 2019.
- Communicated by: Vera Fischer
- © Copyright 2021 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)
- Journal: Proc. Amer. Math. Soc. Ser. B 8 (2021), 267-280
- MSC (2020): Primary 54C35, 54G12, 54H05, 46A03
- DOI: https://doi.org/10.1090/bproc/95
- MathSciNet review: 4316069