Basic properties of X for which the space Cp(X) is distinguished

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

Basic properties of X for which the space Cp(X) is distinguished

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

Basic properties of X for which the space Cp(X) is distinguished

Popis výsledku anglicky

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.

Druh

J_ost - Ostatní články v recenzovaných periodicích

CEP obor

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OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GF20-22230L" target="_blank" >GF20-22230L: Banachovy prostory spojitých a lipschitzovských funkcí</a><br>

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2021

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Proceedings of the American Mathematical Society, Ser. B

ISSN

2330-1511

e-ISSN

2330-1511

Svazek periodika

8

Číslo periodika v rámci svazku

September

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

14

Strana od-do

267-280

Kód UT WoS článku

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EID výsledku v databázi Scopus

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