Efimov spaces and the separable quotient problem for spaces \(C_{p}(K)\). (English) Zbl 1383.46002
The famous separable quotient problem asks whether every infinite-dimensional Banach space admits an infinite-dimensional separable quotient. More generally, the same question can be asked for bigger classes of locally convex spaces instead of the class of Banach spaces. In the recent article [Proc. Am. Math. Soc. 145, No. 9, 3829–3841 (2017; Zbl 1383.46003)], J. Kąkol and S. A. Saxon gave a full characterisation of the completely regular Hausdorff spaces for which the space \(C_p(X)\) of continuous functions on \(X\) equipped with the topology of pointwise convergence admits a separable algebra quotient.
Motivated by the fact that the Stone-Čech compactification \(\beta\mathbb{N}\) of the natural numbers fails these conditions, the current article is devoted to the question of when the space \(C_p(K)\) for an infinite compact space \(K\) admits an infinite-dimensional separable quotient. The main result, giving a sufficient condition for the existence of an infinite-dimensional separable quotient in \(C_p(X)\), reads as follows:
Let \(X\) be a completely regular space with a sequence \((K_n)\) of nonempty compact subsets such that, for each \(n\geq 1\), the set \(K_n\) contains two disjoint subsets homeomorphic to \(K_{n+1}\). Then \(C_p(X)\) has an infinite-dimensional separable quotient. Consequently, if \(K\) is a compact space which contains a copy of \(\beta\mathbb{N}\), then \(C_p(X)\) admits an infinite-dimensional separable quotient.
As corollaries to this result, the authors provide a number of sufficient conditions on \(X\) which ensure that \(C_p(X)\) has such a separable quotient. Moreover, the connection of this result with Efimov spaces is discussed.
The article closes with a list of remarks, examples and problems.
Motivated by the fact that the Stone-Čech compactification \(\beta\mathbb{N}\) of the natural numbers fails these conditions, the current article is devoted to the question of when the space \(C_p(K)\) for an infinite compact space \(K\) admits an infinite-dimensional separable quotient. The main result, giving a sufficient condition for the existence of an infinite-dimensional separable quotient in \(C_p(X)\), reads as follows:
Let \(X\) be a completely regular space with a sequence \((K_n)\) of nonempty compact subsets such that, for each \(n\geq 1\), the set \(K_n\) contains two disjoint subsets homeomorphic to \(K_{n+1}\). Then \(C_p(X)\) has an infinite-dimensional separable quotient. Consequently, if \(K\) is a compact space which contains a copy of \(\beta\mathbb{N}\), then \(C_p(X)\) admits an infinite-dimensional separable quotient.
As corollaries to this result, the authors provide a number of sufficient conditions on \(X\) which ensure that \(C_p(X)\) has such a separable quotient. Moreover, the connection of this result with Efimov spaces is discussed.
The article closes with a list of remarks, examples and problems.
Reviewer: Christian Bargetz (Innsbruck)
MSC:
| 46A08 | Barrelled spaces, bornological spaces |
| 54C35 | Function spaces in general topology |
| 46E10 | Topological linear spaces of continuous, differentiable or analytic functions |
Citations:
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