The Ascoli property for function spaces

0464468 - MÚ 2017 RIV NL eng J - Journal Article
Gabriyelyan, S. - Grebík, Jan - Kąkol, Jerzy - Zdomskyy, L.
The Ascoli property for function spaces.
Topology and its Applications. Roč. 214, 1 December (2016), s. 35-50. ISSN 0166-8641. E-ISSN 1879-3207
R&D Projects: GA ČR(CZ) GF15-34700L; GA ČR GF16-34860L; GA MŠMT(CZ) 7AMB15AT035
Institutional support: RVO:67985840
Keywords : ascoli * paracompact * scattered
Subject RIV: BA - General Mathematics
Impact factor: 0.377, year: 2016
http://www.sciencedirect.com/science/article/pii/S0166864116302152

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.
Permanent Link: http://hdl.handle.net/11104/0263319