TY - JOUR

T1 - BASIC PROPERTIES OF X FOR WHICH THE SPACE Cp(X) IS DISTINGUISHED

AU - Ka¸Kol, Jerzy

AU - Leiderman, Arkady

N1 - Funding Information:
Received by the editors March 23, 2021, and, in revised form, June 15, 2021. 2020 Mathematics Subject Classification. Primary 54C35, 54G12, 54H05, 46A03. Key words and phrases. Distinguished locally convex space, Δ-set, closed mapping, scattered space. The research for the first author was supported by the GACˇR 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.
Funding Information:
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.
Publisher Copyright:
© 2021 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)

PY - 2021/1/1

Y1 - 2021/1/1

N2 - In our paper [Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 86–99] we showed that a Tychonoff space X is a Δ-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 Cp(X) is distinguished. Continuing this research, we investigate whether the class Δ of Δ-spaces is invariant under the basic topological operations. We prove that if X Δ and ϕ : X → Y is a continuous surjection such that ϕ(F ) is an Fσ-set in Y for every closed set F ⊂ X, then also Y Δ. As a consequence, if X is a countable union of closed subspaces Xi such that each Xi Δ, then also X Δ. In particular, σ-product of any family of scattered Eberlein compact spaces is a Δ-space and the product of a Δ-space with a countable space is a Δ-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 : Cp(X) −→ Cp(Y ) be a continuous linear surjection. We observe that T admits an extension to a linear continuous operator T from RX onto RY and deduce that Y is a Δ-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 Δ-space has countable tightness. As a consequence, under Proper Forcing Axiom every compact Δ-space is sequential. In the article we pose a dozen open questions.

AB - In our paper [Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 86–99] we showed that a Tychonoff space X is a Δ-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 Cp(X) is distinguished. Continuing this research, we investigate whether the class Δ of Δ-spaces is invariant under the basic topological operations. We prove that if X Δ and ϕ : X → Y is a continuous surjection such that ϕ(F ) is an Fσ-set in Y for every closed set F ⊂ X, then also Y Δ. As a consequence, if X is a countable union of closed subspaces Xi such that each Xi Δ, then also X Δ. In particular, σ-product of any family of scattered Eberlein compact spaces is a Δ-space and the product of a Δ-space with a countable space is a Δ-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 : Cp(X) −→ Cp(Y ) be a continuous linear surjection. We observe that T admits an extension to a linear continuous operator T from RX onto RY and deduce that Y is a Δ-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 Δ-space has countable tightness. As a consequence, under Proper Forcing Axiom every compact Δ-space is sequential. In the article we pose a dozen open questions.

KW - Distinguished locally convex space

KW - closed mapping

KW - scattered space

KW - Δ-set

UR - http://www.scopus.com/inward/record.url?scp=85164580122&partnerID=8YFLogxK

U2 - 10.1090/bproc/95

DO - 10.1090/bproc/95

M3 - Article

AN - SCOPUS:85164580122

SN - 2330-1511

VL - 8

SP - 267

EP - 280

JO - Proceedings of the American Mathematical Society, Series B

JF - Proceedings of the American Mathematical Society, Series B

ER -