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Filters and sequences. (English) Zbl 0976.03053

If \(F\) is a filter on \(\omega\) let \(C_F\) be the set of all sequences \(\{x_n\}\) of real numbers which have limit with respect to \(F\), i.e. such that for some real \(r\) we have \(\{n : |x_n - r|< \varepsilon\} \in F\) for all \(\varepsilon > 0\). T. Dobrowolski and W. Marciszewski [Fundam. Math. 148, 35-62 (1995; Zbl 0834.46016)] proved that if \(F\) is \(\Pi^0_\alpha\) then \(C_F\) is \(\Pi^0_\alpha\)-hard and is the difference of two \(\Pi^0_\alpha\) sets. In the first section of the paper the gap between these bounds on the complexity of \(C_F\) is closed when \(\alpha = 3\): if \(F\) is \(\Pi^0_3\) then \(C_F\) is \(\Pi^0_3\) (and thus, by another result of Dobrowolski and Marciszewski, all \(C_F\)’s with \(F \in \Pi^0_3\) are homeomorphic).
The second section of the paper deals with filters such that Fatou’s lemma holds when limits are taken with respect to the filter (these filters have been studied by A. Louveau [Ann. Inst. Fourier 36, 57-68 (1986; Zbl 0604.46012)]). Solecki defines a filter \(F_0\) such that a filter satisfies Fatou’s lemma if and only if it does not locally contain \(F_0\).

MSC:

03E15 Descriptive set theory
46E10 Topological linear spaces of continuous, differentiable or analytic functions
54C35 Function spaces in general topology
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
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