2017/09/06 by Balcerzak, Marek, Leonetti, Paolo · 1 citation
#11B05 #40A05 #40A35 #54A20 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN) #Number Theory (math.NT) #Probability (math.PR)
paper · doi:10.48550/arxiv.1709.01680
Let X be a first countable space which admits a non-trivial convergent sequence and let I be an analytic P-ideal. First, it is shown that the sets of I-limit points of all sequences in X are closed if and only if I is also an Fσ-ideal. Moreover, let (xn) be a sequence taking values in a Polish space without isolated points. It is known that the set A of its statistical limit points is an Fσ-set, the set B of its statistical cluster points is closed, and that the set C of its ordinary limit points is closed, with A⊆ B⊆ C. It is proved the sets A and B own some additional relationship: indeed, the set S of isolated points of B is contained also in A. Conversely, if A is an Fσ-set, B is a closed set with a subset S of isolated points such that B∖ S≠ ∅ is regular closed, and C is a closed set with S⊆ A⊆ B⊆ C, then there exists a sequence (xn) for which: A is the set of its statistical limit points, B is the set of its statistical cluster points, and C is the set of its ordinary limit points. Lastly, we discuss topological nature of the set of I-limit points when I is neither Fσ- nor analytic P-ideal.