A characterization of X for which spaces C_p(X) are distinguished and applications
Result description
We prove that the locally convex space of continuous real-valued functions on a Tychonoff space equipped with the topology of pointwise convergence is distinguished if and only if is a -space in the sense of Knight in [Trans. Amer. Math. Soc. 339 (1993), pp. 45–60]. As an application of this characterization theorem we obtain the following results: [1)] If is a Čech-complete (in particular, compact) space such that is distinguished, then is scattered. [2)] For every separable compact space of the Isbell–Mrówka type , the space is distinguished. [3)] If is the compact space of ordinals , then is not distinguished. We observe that the existence of an uncountable separable metrizable space such that is distinguished, is independent of ZFC. We also explore the question to which extent the class of -spaces is invariant under basic topological operations.
Keywords
distinguished locally convex spacescattered compact spacedelta-setIsbell–Mrówka space
The result's identifiers
Result code in IS VaVaI
Result on the web
DOI - Digital Object Identifier
Alternative languages
Result language
angličtina
Original language name
A characterization of X for which spaces C_p(X) are distinguished and applications
Original language description
We prove that the locally convex space of continuous real-valued functions on a Tychonoff space equipped with the topology of pointwise convergence is distinguished if and only if is a -space in the sense of Knight in [Trans. Amer. Math. Soc. 339 (1993), pp. 45–60]. As an application of this characterization theorem we obtain the following results: [1)] If is a Čech-complete (in particular, compact) space such that is distinguished, then is scattered. [2)] For every separable compact space of the Isbell–Mrówka type , the space is distinguished. [3)] If is the compact space of ordinals , then is not distinguished. We observe that the existence of an uncountable separable metrizable space such that is distinguished, is independent of ZFC. We also explore the question to which extent the class of -spaces is invariant under basic topological operations.
Czech name
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Czech description
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Classification
Type
Jost - Miscellaneous article in a specialist periodical
CEP classification
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OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
GF20-22230L: Banach spaces of continuous and Lipschitz functions
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
February
Country of publishing house
US - UNITED STATES
Number of pages
14
Pages from-to
86-99
UT code for WoS article
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EID of the result in the Scopus database
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Basic information
Result type
Jost - Miscellaneous article in a specialist periodical
OECD FORD
Pure mathematics
Year of implementation
2021