Metrizable quotients of Cp-spaces

0493785 - MÚ 2019 RIV NL eng J - Článek v odborném periodiku
Banakh, T. - Kąkol, Jerzy - Śliwa, W.
Metrizable quotients of Cp-spaces.
Topology and its Applications. Roč. 249, 1 November (2018), s. 95-102. ISSN 0166-8641. E-ISSN 1879-3207
Grant CEP: GA ČR GF16-34860L
Institucionální podpora: RVO:67985840
Klíčová slova: Efimov space * function space * metrizable quotient * pseudocompact space
Obor OECD: Pure mathematics
Impakt faktor: 0.416, rok: 2018
https://www.sciencedirect.com/science/article/pii/S0166864118303699?via%3Dihub

The famous Rosenthal–Lacey theorem asserts that for each infinite compact set K the Banach space C(K) admits a quotient which is either an isomorphic copy of c or ℓ2. What is the case when the uniform topology of C(K) is replaced by the pointwise topology? Is it true that Cp(X) always has an infinite-dimensional separable (or better metrizable) quotient? In this paper we prove that for a Tychonoff space X the function space Cp(X) has an infinite-dimensional metrizable quotient if X either contains an infinite discrete C⁎-embedded subspace or else X has a sequence (Kn)n\in N of infinite compact subsets such that for every n the space Kn contains two disjoint topological copies of Kn+1. Applying the latter result, we show that under ◊ there exists a zero-dimensional Efimov space K whose function space Cp(K) has an infinite-dimensional metrizable quotient. These two theorems essentially improve earlier results of Ka̧kol and Śliwa on infinite-dimensional separable quotients of Cp-spaces.
Trvalý link: http://hdl.handle.net/11104/0287092