@MISC{CARDFIL4.MIZ,
    TITLE = {Double Sequences and Iterated Limits in Regular Space},
    AUTHOR = {Coghetto, Roland},
    ADDRESS1 = {Rue de la Brasserie 5\\ 7100 La Louvi\`ere, Belgium},
    DAY = {30},
    MONTH = {June},
    YEAR = {2016},
    SUMMARY = {First, we define the Cartesian product of two filters bases and the
  cartesian product of two filters. \par
  After comparing the product of two Frechet filters on
  $\mathbb N$ (${\cal F}_1$) with the Frechet filter on
  $\mathbb N\times \mathbb N$ (${\cal F}_2$), we compare $lim_{{\cal F}_1}$
  and $\lim{{\cal F}_2}$ for all double sequences in a non empty
  topological space. \par
  Endou, Okazaki, and Shidama formalized
  the ``convergence in Pringsheim's sense'' \cite{DBLSEQ_1.ABS}
  for double sequence of real numbers. We show some basic correspondences
  between the p-convergence and the filter convergence
  in a topological space. \par
  After we formalize that the double sequence
  $(x_{m,n}=\frac{1}{m+1})_{(m,n)\in{\mathbb{N}\times\mathbb{N}}$ converges
  in ``Pringsheim's sense" but not in Frechet filter on
  $\mathbb{N}\times\mathbb{N}$" sense.
  \par
  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" \cite{DBLSEQ_1.ABS}
  in Hausdorff space.
  \par
  Finally, we generalize two theorems: 3 and 4 from \cite{DBLSEQ_1.ABS}
  in the case of
  double sequences and we formalize the ``iterated limit" theorem
  (``Double limit" \cite{bourbaki2013general}, p. 81, par. 8.5
  \textit{``Double limite"} \cite{bourbaki2007topologie} (TG I,57)),
  all in regular space.
  \par
  We are inspired by the exercises (2.11.4), (2.17.5) \cite{wagschal} and
  the corrections B.10 \cite{wagschaltopex}.},
    MSC2010 = {54A20; 40A05; 40B05; 03B35},
    KEYWORDS = {Filter, Double limits, Pringsheim convergence,
    Iterated limits, Regular space}}

@book{wagschal,
title={Topologie et analyse fonctionnelle},
author={Wagschal, Claude},
publisher={Hermann},
year={1995}}

@book{wagschaltopex,
title={Topologie: Exercices et problèmes corrig\´es},
author={Wagschal, Claude},
publisher={Hermann},
year={1995}}

@book{bourbaki2007topologie,
  title={Topologie g{\'e}n{\'e}rale: chapitres 1 {\`a} 4},
  author={Bourbaki, Nicolas},
  year={2007},
  publisher={Springer Science \& Business Media}}

@book{bourbaki2013general,
  title={General Topology: Chapters 1--4},
  author={Bourbaki, Nicolas},
  year={2013},
  publisher={Springer Science and Business Media}}
