Metrizable quotients of $C_p$-spaces
| Autor: | Banakh, T., Kąkol, J., Śliwa, W. |
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| Rok vydání: | 2018 |
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| Zdroj: | Topology Appl. 249:1 (2018) 95-102 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1016/j.topol.2018.09.012 |
| Popis: | The famous Rosenthal-Lacey theorem asserts that for each infinite compact set $K$ the Banach space $C(K)$ admits a quotient which is either a copy of $c$ or $\ell_{2}$. What is the case when the uniform topology of $C(K)$ is replaced by the pointwise topology? Is it true that $C_p(X)$ always has an infinite-dimensional separable (or better metrizable) quotient? In this paper we prove that for a Tychonoff space $X$ the function space $C_p(X)$ has an infinite-dimensional metrizable quotient if $X$ either contains an infinite discrete $C^*$-embedded subspace or else $X$ has a sequence $(K_n)_{n\in\mathbb N}$ of compact subsets such that for every $n$ the space $K_n$ contains two disjoint topological copies of $K_{n+1}$. Applying the latter result, we show that under $\lozenge$ there exists a zero-dimensional Efimov space $K$ whose function space $C_{p}(K)$ has an infinite-dimensional metrizable quotient. These two theorems essentially improve earlier results of K\k{a}kol and \'Sliwa on infinite-dimensional separable quotients of $C_p$-spaces. Comment: 8 pages |
| Databáze: | arXiv |
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