Let X be a first countable space which has a non-isolated point 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. 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≠∅, 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.

On the Relationship between Ideal Cluster Points and Ideal Limit Points

Leonetti, Paolo
2019-01-01

Abstract

Let X be a first countable space which has a non-isolated point 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. 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≠∅, 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.
2019
2018
Analytic P-ideal; Asymptotic density; Co-analytic ideal; Equidistribution; Ideal cluster point; Ideal limit point; Maximal ideal; Regular closed set
Balcerzak, Mareka; Leonetti, Paolo
File in questo prodotto:
File Dimensione Formato  
PaoloMarek_2018_11_29_final.pdf

accesso aperto

Tipologia: Documento in Post-print
Licenza: Creative commons
Dimensione 490.85 kB
Formato Adobe PDF
490.85 kB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/2142080
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 15
  • ???jsp.display-item.citation.isi??? 15
social impact