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.


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