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Double Sequences and Iterated Limits in Regular Space

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Coghetto R.

Formalized Mathematics

2016

tom 24

nr 3

173-186

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].

Ograniczanie wyników

1 Formalized Mathematics

1 Coghetto R.

1 2016

Wyniki wyszukiwania

Double Sequences and Iterated Limits in Regular Space