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Separable (and metrizable) infinite dimensional quotients of Cp(X) and Cc(X) spaces
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SYSNO ASEP 0505923 Document Type C - Proceedings Paper (int. conf.) R&D Document Type Conference Paper Title Separable (and metrizable) infinite dimensional quotients of Cp(X) and Cc(X) spaces Author(s) Kąkol, Jerzy (MU-W) SAI, RID, ORCID Source Title Descriptive Topology and Functional Analysis II. - Cham : Springer, 2019 / Ferrando J. C. - ISSN 2194-1009 - ISBN 978-3-030-17375-3 Pages s. 175-189 Number of pages 15 s. Publication form Print - P Action 2nd Meeting in Topology and Functional Analysis, In Honour of Manuel López-Pellicer Mathematical Work Event date 07.06.2018 - 08.06.2018 VEvent location Elche Country ES - Spain Event type WRD Language eng - English Country CH - Switzerland Keywords the separable quotient problem ; spaces of continuous functions ; quotient spaces ; the Josefson-Nissenzweig theorem ; Efimov space Subject RIV BA - General Mathematics OECD category Pure mathematics R&D Projects GF16-34860L GA ČR - Czech Science Foundation (CSF) Institutional support MU-W - RVO:67985840 EID SCOPUS 85067354596 DOI 10.1007/978-3-030-17376-0_10 Annotation The famous Rosenthal-Lacey theorem states that for each infinite compact set K the Banach space C(K) of continuous real-valued functions on a compact space K admits a quotient which is either an isomorphic copy of c or ℓ2. Whether C(K) admits an infinite dimensional separable (or even metrizable) Hausdorff quotient when the uniform topology of C(K) is replaced by the pointwise topology remains as an open question. The present survey paper gathers several results concerning this question for the space Cp(K) of continuous real-valued functions endowed with the pointwise topology. Among others, that Cp(K) has an infinite dimensional separable quotient for any compact space K containing a opy of βN. Consequently, this result reduces the above question to the case when K is a Efimov space (i.e. K is an infinite compact space that contains neither a non-trivial convergent sequence nor a copy of βN). On the other hand, although it is unknown if Efimov spaces exist in ZFC, we note under (applying some result due to R. de la Vega), that for some Efimov space K the space Cp(K) has an infinite dimensional (even metrizable) separable quotient. The last part discusses the so-called Josefson–Nissenzweig property for spaces Cp(K), introduced recently in [3], and its relation with the separable quotient problem for spaces Cp(K). Workplace Mathematical Institute Contact Jarmila Štruncová, struncova@math.cas.cz, library@math.cas.cz, Tel.: 222 090 757 Year of Publishing 2020
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