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The Ascoli property for function spaces
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SYSNO ASEP 0464468 Document Type J - Journal Article R&D Document Type Journal Article Subsidiary J Článek ve WOS Title The Ascoli property for function spaces Author(s) Gabriyelyan, S. (IL)
Grebík, Jan (MU-W) SAI, RID
Kąkol, Jerzy (MU-W) SAI, RID, ORCID
Zdomskyy, L. (AT)Source Title Topology and its Applications. - : Elsevier - ISSN 0166-8641
Roč. 214, 1 December (2016), s. 35-50Number of pages 16 s. Language eng - English Country NL - Netherlands Keywords ascoli ; paracompact ; scattered Subject RIV BA - General Mathematics R&D Projects GF15-34700L GA ČR - Czech Science Foundation (CSF) GF16-34860L GA ČR - Czech Science Foundation (CSF) 7AMB15AT035 GA MŠMT - Ministry of Education, Youth and Sports (MEYS) Institutional support MU-W - RVO:67985840 UT WOS 000389391700003 EID SCOPUS 84989181029 DOI 10.1016/j.topol.2016.08.026 Annotation The paper deals with Ascoli spaces Cp(X) and Ck(X) over Tychonoff spaces X. The class of Ascoli spaces X, i.e. spaces X for which any compact subset K of Ck(X) is evenly continuous, essentially includes the class of kR-spaces. First we prove that if Cp(X) is Ascoli, then it is κ-Fréchet–Urysohn. If X is cosmic, then Cp(X) is Ascoli iff it is κ-Fréchet–Urysohn. This leads to the following extension of a result of Morishita: If for a Čech-complete space X the space Cp(X) is Ascoli, then X is scattered. If X is scattered and stratifiable, then Cp(X) is an Ascoli space. Consequently: (a) If X is a complete metrizable space, then Cp(X) is Ascoli iff X is scattered. (b) If X is a Čech-complete Lindelöf space, then Cp(X) is Ascoli iff X is scattered iff Cp(X) is Fréchet–Urysohn. Moreover, we prove that for a paracompact space X of point-countable type the following conditions are equivalent: (i) X is locally compact. (ii) Ck(X) is a kR-space. (iii) Ck(X) is an Ascoli space. Workplace Mathematical Institute Contact Jarmila Štruncová, struncova@math.cas.cz, library@math.cas.cz, Tel.: 222 090 757 Year of Publishing 2017
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