Separable quotients for less-than-barrelled function spaces. (English) Zbl 1387.46023
Let \(X\) be a completely regular Hausdorff topological space. Denote by \(C_c(X)\) and \(C_p(X)\) the linear space \(C(X)\) of all continuous real-valued functions endowed with the pointwise and compact-open topologies, respectively. This article addresses the problem of finding (always Hausdorff infinite dimensional) separable quotients of \(C_c(X)\) and \(C_p(X)\). The authors and A. R. Todd proved in [Bull. Aust. Math. Soc. 90, No. 2, 295–303 (2014; Zbl 1306.46002)] that every barrelled \(C_c(X)\) admits a separable quotient. In the paper under review, it is shown that, if \(C_c(X)\) (resp., \(C_p(X)\)) is dual locally complete (a much weaker condition than being barrelled), then this space has a separable quotient. Examples of spaces of this type which are not dual locally complete but have a separable quotient are presented.
Reviewer: José Bonet (Valencia)
MSC:
| 46E10 | Topological linear spaces of continuous, differentiable or analytic functions |
| 46A08 | Barrelled spaces, bornological spaces |
| 54C35 | Function spaces in general topology |
Citations:
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