On the relationship between ideal cluster points and ideal limit points

Balcerzak, Marek; Leonetti, Paolo

doi:10.1016/j.topol.2018.11.022

Cited by 22 publications

(16 citation statements)

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“…We refer the reader to [26] for characterizations of I-cluster points and [2] for their relation with I-limit points. Lastly, we recall that the sequence x is said to be I-convergent to ∈ X, shortened as…”

Section: Introductionmentioning

confidence: 99%

The Baire Category of Subsequences and Permutations which preserve Limit Points

Balcerzak

Leonetti

2020

Results Math

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Let $$\mathcal {I}$$ I be a meager ideal on $$\mathbf {N}$$ N . We show that if x is a sequence with values in a separable metric space then the set of subsequences [resp. permutations] of x which preserve the set of $$\mathcal {I}$$ I -cluster points of x is topologically large if and only if every ordinary limit point of x is also an $$\mathcal {I}$$ I -cluster point of x. The analogue statement fails for all maximal ideals. This extends the main results in [Topology Appl. 263 (2019), 221–229]. As an application, if x is a sequence with values in a first countable compact space which is $$\mathcal {I}$$ I -convergent to $$\ell $$ ℓ , then the set of subsequences [resp. permutations] which are $$\mathcal {I}$$ I -convergent to $$\ell $$ ℓ is topologically large if and only if x is convergent to $$\ell $$ ℓ in the ordinary sense. Analogous results hold for $$\mathcal {I}$$ I -limit points, provided $$\mathcal {I}$$ I is an analytic P-ideal.

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Section: Introductionmentioning

confidence: 99%

The Baire Category of Subsequences and Permutations which preserve Limit Points

Balcerzak

Leonetti

2020

Results Math

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“…In particular, Γ x (I) is compact for all real bounded sequences x, cf. [2,13,14] for basic facts and characterizations of I-cluster points.…”

Section: Introductionmentioning

confidence: 99%

Characterizations of the ideal core

Leonetti

2019

Journal of Mathematical Analysis and Applications

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Given an ideal I on ω and a sequence x in a topological vector space, we let the I-core of x be the least closed convex set containing {x n : n / ∈ I} for all I ∈ I. We show two characterizations of the I-core. This implies that the I-core of a bounded sequence in R k is simply the convex hull of its I-cluster points. As applications, we simplify and extend several results in the context of Pringsheim-convergence and e-convergence of double sequences.2010 Mathematics Subject Classification. Primary: 40A35. Secondary: 54A20, 40A05.

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“…The statistical versions of the result appeared in Theorem 1.1 [27] for real sequences, in Theorem 2.6 [28] in topological spaces and then in Theorem 2 [25] from where the statement has been reproduced. Later with a different line of proof it appeared in Theorem 3.1 [29] for metric spaces and in Theorem 2.2 [30] (topological spaces). The converse of the above result was established in Theorem 3 [25] but there was a gap in the argument which was later modified and is given a little later.…”

Section: Theorem 4 ([21]mentioning

confidence: 99%