A new combinatorial characterization of the minimal cardinality of a subset of R which is not of first category

Details A new combinatorial characterization of the minimal cardinality of a subset of R which is not of first category A new combinatorial characterization of the minimal cardinality of a subset of R which is not of first category

1999-11-17

arxiv.org/abs/math/9911122v2

Mathematics

Logic

4 pages, LaTeX 209, a new theorem added (see Abstract and Note on p.2)

Scientific paper

Let M denote the ideal of first category subsets of R. We prove that min{card X: X \subseteq R, X \not\in M} is the smallest cardinality of a family S \subseteq {0,1}^\omega with the property that for each f: \omega -> \bigcup_{n \in \omega}{0,1}^n there exists a sequence {a_n}_{n \in \omega} belonging to S such that for infinitely many i \in \omega the infinite sequence {a_{i+n}}_{n \in \omega} extends the finite sequence f(i). We inform that S \subseteq {0,1}^\omega is not of first category if and only if for each f: \omega -> \bigcup_{n \in \omega}{0,1}^n there exists a sequence {a_n}_{n \in \omega} belonging to S such that for infinitely many i \in \omega the infinite sequence {a_{i+n}}_{n \in \omega} extends the finite sequence f(i).

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