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On the Egoroff property of pointwise convergent sequences of functions. (English) Zbl 0601.54004

The space \({\mathcal L}(X)\) of real-valued functions on X has the Egoroff property if for any \(\{f_{nk}\}\) such that \(0\leq f_{nk}\uparrow_ kf\) (for every n), there exists \(g_ m\uparrow f\) such that, for each m and n, \(g_ m\leq f_{nk}\) for some k. We show that \({\mathcal L}(X)\) has the Egoroff property if and only if the cardinality of X is smaller than the minimum cardinality of an unbounded family of functions from the set of natural numbers to itself. Therefore, the statement that there is an uncountable set X such that \({\mathcal L}(X)\) has the Egoroff property is independent of the axioms of set theory.

MSC:

54A35 Consistency and independence results in general topology
54C35 Function spaces in general topology
03E05 Other combinatorial set theory
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References:

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