Double Sequences and Iterated Limits in Regular Space

• Autorzy: Roland Coghetto
• Języki publikacji: EN

Abstrakty

EN

First, we define in Mizar [5], the Cartesian product of two filters bases and the Cartesian product of two filters. After comparing the product of two Fréchet filters on ℕ (F1) with the Fréchet filter on ℕ × ℕ (F2), we compare limF₁ and limF₂ for all double sequences in a non empty topological space. Endou, Okazaki and Shidama formalized in [14] the “convergence in Pringsheim’s sense” for double sequence of real numbers. We show some basic correspondences between the p-convergence and the filter convergence in a topological space. Then we formalize that the double sequence [...] converges in “Pringsheim’s sense” but not in Frechet filter on ℕ × ℕ sense. In the next section, we generalize some definitions: “is convergent in the first coordinate”, “is convergent in the second coordinate”, “the lim in the first coordinate of”, “the lim in the second coordinate of” according to [14], in Hausdorff space. Finally, we generalize two theorems: (3) and (4) from [14] in the case of double sequences and we formalize the “iterated limit” theorem (“Double limit” [7], p. 81, par. 8.5 “Double limite” [6] (TG I,57)), all in regular space. We were inspired by the exercises (2.11.4), (2.17.5) [17] and the corrections B.10 [18].

Słowa kluczowe

EN

filter; double limits; Pringsheim convergence; iterated limits; regular space

• Wydawca: De Gruyter Open
• Czasopismo: Formalized Mathematics
• Rocznik: 2016
• Tom: 24
• Numer: 3
• Strony: 173-186

Daty

wydano

2016-09-01

otrzymano

2016-06-30

online

2017-02-08

Twórcy

autor

Roland Coghetto

• Rue de la Brasserie 5 7100 La Louvière,

Identyfikatory

DOI

10.1515/forma-2016-0014