A new characterization of compact scattered spaces X in terms of spaces Cp(X)

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F25%3A00617512" target="_blank" >RIV/67985840:_____/25:00617512 - isvavai.cz</a>

Result on the web

<a href="https://doi.org/10.1007/s13398-025-01700-9" target="_blank" >https://doi.org/10.1007/s13398-025-01700-9</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s13398-025-01700-9" target="_blank" >10.1007/s13398-025-01700-9</a>

Result language

angličtina

Original language name

A new characterization of compact scattered spaces X in terms of spaces Cp(X)

Original language description

For a Tychonoff space X by Cp(X) we denote the space of continuous real valued functions on X endowed with the pointwise topology, and C(X) denotes the Banach space endowed with the uniform topology provided X is compact. The classical two results characterizing compact scattered spaces in terms of C(X) and Cp(X) assert that a compact space X is scattered if and only if C(X) is an Asplund space (Namioka-Phelps) if and only if Cp(X) is a Fréchet-Urysohn space (Gerlits, Pytkeev). We provide another result of this type by showing the following Theorem: An infinite compact space X is scattered if and only if Cp(X) contains no closed Q-compact infinite-dimensional vector subspace if and only if Cp(X) contains no infinite-dimensional vector subspace admitting a fundamental sequence of bounded sets if and only if every vector subspace of Cp(X) is bornological. The above Theorem fails if X is nondiscrete scattered and noncompact. On the other hand, if X is a countable metric space which is not scattered, Cp(X) contains a closed infinite-dimensional Q-compact subspace. Moreover, if X = F x [1, w] and F is discrete with F >= d, where d is the dominating cardinal, then Cp(X) contains a closed infinite-dimensional Q-compact subspace, but if X = N x [1, w] the corresponding space C-p(X) does not contain such subspaces. A variant of Theorem is also obtained characterizing infinite Tychonoff spaces X for which all compact subsets are scattered. These results are also motivated by a remarkable theorem of Velichko stating that for an infinite Tychonoff space X the space Cp(X) is not Q-compact. Several illustrating examples involving spaces c(0), ( pound infinity) and the space Lip(0)(M) with the pointwise topology are provided and discussed.

Czech name

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Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

—

OECD FORD branch

10101 - Pure mathematics

Project

<a href="/en/project/GF22-07833K" target="_blank" >GF22-07833K: Homogeneity and Genericity of Metric Structures - Groups, Dynamical Systems, Banach Spaces and C*-Algebras</a><br>

Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2025

Confidentiality

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

Name of the periodical

Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales

ISSN

1578-7303

e-ISSN

1579-1505

Volume of the periodical

119

Issue of the periodical within the volume

2

Country of publishing house

DE - GERMANY

Number of pages

16

Pages from-to

43

UT code for WoS article

001419846600002

EID of the result in the Scopus database

2-s2.0-85218352605