Basic properties of X for which the space Cp(X) is distinguished
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F21%3A00549289" target="_blank" >RIV/67985840:_____/21:00549289 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.1090/bproc/95" target="_blank" >https://doi.org/10.1090/bproc/95</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1090/bproc/95" target="_blank" >10.1090/bproc/95</a>
Alternative languages
Result language
angličtina
Original language name
Basic properties of X for which the space Cp(X) is distinguished
Original language description
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.
Czech name
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Czech description
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Classification
Type
J<sub>ost</sub> - Miscellaneous article in a specialist periodical
CEP classification
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OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
<a href="/en/project/GF20-22230L" target="_blank" >GF20-22230L: Banach spaces of continuous and Lipschitz functions</a><br>
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2021
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Proceedings of the American Mathematical Society, Ser. B
ISSN
2330-1511
e-ISSN
2330-1511
Volume of the periodical
8
Issue of the periodical within the volume
September
Country of publishing house
US - UNITED STATES
Number of pages
14
Pages from-to
267-280
UT code for WoS article
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EID of the result in the Scopus database
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