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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

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1898-9934
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In This Section
Volume 24, Issue 3 (Sep 2016)

Issues

Double Sequences and Iterated Limits in Regular Space

Roland Coghetto
  • Rue de la Brasserie 5 7100 La Louvière, Belgium
Published Online: 2017-02-21 | DOI: https://doi.org/10.1515/forma-2016-0014

Abstract

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].

MSC: 54A20; 40A05; 40B05; 03B35

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

MML: identifier: CARDFIL4; version: 8.1.05 5.37.1275

References

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About the article

Received: 2016-06-30

Published Online: 2017-02-21

Published in Print: 2016-09-01



Citation Information: Formalized Mathematics, ISSN (Online) 1898-9934, DOI: https://doi.org/10.1515/forma-2016-0014. Export Citation

© by Roland Coghetto. This work is licensed under version 3.0 of the Creative Commons Attribution–ShareAlike License (CC BY-SA 3.0 LEGALCODE)

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