Limit points of subsequences

2017/12/31 by Leonetti, Paolo · 1 citation
#54A20 #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN) #Primary: 40A35. Secondary: 11B05 #Probability (math.PR)

paper · doi:10.48550/arxiv.1801.00343

Abstract

Let x be a sequence taking values in a separable metric space and I be a generalized density ideal or an F_σ-ideal on the positive integers (in particular, I can be any Erd\H os--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 [respectively, I-limit points] is of second category if and only if the set of I-cluster points of x [resp., I-limit points] coincides with the set of ordinary limit points of x; moreover, in this case, it is comeager. In particular, it follows that the collection of subsequences of x which preserve the set of ordinary limit points of x is comeager.

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• On the Relationship between Ideal Cluster Points and Ideal Limit Points
• Characterizations of Ideal Cluster Points