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Separable (and metrizable) infinite dimensional quotients of Cp(X) and Cc(X) spaces
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SYSNO ASEP 0505923 Druh ASEP C - Konferenční příspěvek (mezinárodní konf.) Zařazení RIV D - Článek ve sborníku Název Separable (and metrizable) infinite dimensional quotients of Cp(X) and Cc(X) spaces Tvůrce(i) Kąkol, Jerzy (MU-W) SAI, RID, ORCID Zdroj.dok. Descriptive Topology and Functional Analysis II. - Cham : Springer, 2019 / Ferrando J. C. - ISSN 2194-1009 - ISBN 978-3-030-17375-3 Rozsah stran s. 175-189 Poč.str. 15 s. Forma vydání Tištěná - P Akce 2nd Meeting in Topology and Functional Analysis, In Honour of Manuel López-Pellicer Mathematical Work Datum konání 07.06.2018 - 08.06.2018 Místo konání Elche Země ES - Španělsko Typ akce WRD Jazyk dok. eng - angličtina Země vyd. CH - Švýcarsko Klíč. slova the separable quotient problem ; spaces of continuous functions ; quotient spaces ; the Josefson-Nissenzweig theorem ; Efimov space Vědní obor RIV BA - Obecná matematika Obor OECD Pure mathematics CEP GF16-34860L GA ČR - Grantová agentura ČR Institucionální podpora MU-W - RVO:67985840 EID SCOPUS 85067354596 DOI 10.1007/978-3-030-17376-0_10 Anotace 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). Pracoviště Matematický ústav Kontakt Jarmila Štruncová, struncova@math.cas.cz, library@math.cas.cz, Tel.: 222 090 757 Rok sběru 2020
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