We show that an ideal I on the positive integers is meager if and only if there exists a bounded nonconvergent real sequence x such that the set of subsequences [resp. permutations] of x which preserve the set of I-limit points is comeager and, in addition, every accumulation point of x is also an I-limit point (that is, a limit of a subsequence (x n k ) such that {n1, n2, . . . , } /∈ I). The analogous characterization holds also for I-cluster points.
Another characterization of meager ideals
Paolo Leonetti
2023-01-01
Abstract
We show that an ideal I on the positive integers is meager if and only if there exists a bounded nonconvergent real sequence x such that the set of subsequences [resp. permutations] of x which preserve the set of I-limit points is comeager and, in addition, every accumulation point of x is also an I-limit point (that is, a limit of a subsequence (x n k ) such that {n1, n2, . . . , } /∈ I). The analogous characterization holds also for I-cluster points.
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