Let x be a sequence taking values in a separable metric space and let I be an Fσδ-ideal on the positive integers (in particular, I can be any Erdős–Ulam ideal or any summable ideal). It is shown that the collection of subsequences of x which preserve the set of I-cluster points of x is of second category if and only if the set of I-cluster points of x coincides with the set of ordinary limit points of x; moreover, in this case, it is comeager. The analogue for I-limit points is provided. As a consequence, the collection of subsequences of x which preserve the set of ordinary limit points is comeager.

Limit points of subsequences

Leonetti P
2019

Abstract

Let x be a sequence taking values in a separable metric space and let I be an Fσδ-ideal on the positive integers (in particular, I can be any Erdős–Ulam ideal or any summable ideal). It is shown that the collection of subsequences of x which preserve the set of I-cluster points of x is of second category if and only if the set of I-cluster points of x coincides with the set of ordinary limit points of x; moreover, in this case, it is comeager. The analogue for I-limit points is provided. As a consequence, the collection of subsequences of x which preserve the set of ordinary limit points is comeager.
Asymptotic density; Erdős–Ulam ideal; Fσ-ideal; Generalized density ideal; Ideal cluster points; Ideal limit points; Meager set; Summable ideal
Leonetti, P
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/2142096
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