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Topological properties of function spaces: Duality theorems. (English. Russian original) Zbl 0586.54021

Sov. Math., Dokl. 27, 470-473 (1983); translation from Dokl. Akad. Nauk SSSR 269, 1289-1291 (1983).
Let \(C_ p(X)\) denote the space of all continuous, real-valued functions defined on a Tikhonov space X with the topology of pointwise convergence. A space X is said to be \(\tau\)-simple for some infinite cardinal \(\tau\) in case X has the property that for each continuous mapping f from X into a space of weight \(\leq \tau\), the cardinality of f(X) is \(\leq \tau\). A space which is \(\tau\)-simple for all infinite cardinals \(\tau\) is said to be simple. A space X is said to be strongly \(\tau\)-monolithic in case the closure of each subset with cardinality \(\leq \tau\) has weight \(\leq \tau\). If X is strongly \(\tau\)-monolithic for all infinite cardinals \(\tau\), then X is said to be strongly monolithic. The author’s main result says that a space X is \(\tau\)-simple (simple) if and only if the space \(C_ p(X)\) is strongly \(\tau\)-monolithic (strongly monolithic). He states a number of other results relating these and similar topological notions based on the weight and Lindelöf number of space, and lists some open questions. He also cites some related results of other authors, including an example due to E. G. Pytkeev of spaces X and Y such that X is strongly monolithic, Y is not strongly \(\aleph_ 0\)-monolithic, but \(C_ p(X)\) and \(C_ p(Y)\) are linearly homeomorphic.
Reviewer: J.V.Whittaker

MSC:

54C35 Function spaces in general topology
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
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