Journal of Formalized Mathematics
Volume 10, 1998
University of Bialystok
Copyright (c) 1998 Association of Mizar Users

First-countable, Sequential, and Frechet Spaces


Bartlomiej Skorulski
University of Bialystok

Summary.

This article contains a definition of three classes of topological spaces: first-countable, Frechet, and sequential. Next there are some facts about them, that every first-countable space is Frechet and every Frechet space is sequential. Next section contains a formalized construction of topological space which is Frechet but not first-countable. This article is based on [10, pp. 73-81].

MML Identifier: FRECHET

The terminology and notation used in this paper have been introduced in the following articles [18] [21] [20] [11] [1] [13] [22] [5] [6] [17] [2] [3] [7] [16] [14] [8] [12] [4] [9] [15] [19]

Contents (PDF format)

  1. Preliminaries
  2. First-countable, Sequential, and {F}rechet Spaces
  3. Counterexample of {F}rechet but Not First-countable Space
  4. Auxiliary Theorems

Bibliography

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Journal of Formalized Mathematics, 1, 1989.
[2] Grzegorz Bancerek. The ordinal numbers. Journal of Formalized Mathematics, 1, 1989.
[3] Grzegorz Bancerek. Countable sets and Hessenberg's theorem. Journal of Formalized Mathematics, 2, 1990.
[4] Leszek Borys. Paracompact and metrizable spaces. Journal of Formalized Mathematics, 3, 1991.
[5] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[6] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[7] Czeslaw Bylinski. The modification of a function by a function and the iteration of the composition of a function. Journal of Formalized Mathematics, 2, 1990.
[8] Agata Darmochwal. Families of subsets, subspaces and mappings in topological spaces. Journal of Formalized Mathematics, 1, 1989.
[9] Agata Darmochwal and Yatsuka Nakamura. Metric spaces as topological spaces --- fundamental concepts. Journal of Formalized Mathematics, 3, 1991.
[10] Ryszard Engelking. \em General Topology, volume 60 of \em Monografie Matematyczne. PWN - Polish Scientific Publishers, Warsaw, 1977.
[11] Krzysztof Hryniewiecki. Basic properties of real numbers. Journal of Formalized Mathematics, 1, 1989.
[12] Stanislawa Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Journal of Formalized Mathematics, 2, 1990.
[13] Beata Padlewska. Families of sets. Journal of Formalized Mathematics, 1, 1989.
[14] Beata Padlewska and Agata Darmochwal. Topological spaces and continuous functions. Journal of Formalized Mathematics, 1, 1989.
[15] Jan Popiolek. Real normed space. Journal of Formalized Mathematics, 2, 1990.
[16] Konrad Raczkowski and Pawel Sadowski. Topological properties of subsets in real numbers. Journal of Formalized Mathematics, 2, 1990.
[17] Andrzej Trybulec. Binary operations applied to functions. Journal of Formalized Mathematics, 1, 1989.
[18] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[19] Andrzej Trybulec. Baire spaces, Sober spaces. Journal of Formalized Mathematics, 9, 1997.
[20] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.
[21] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[22] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.

Received May 13, 1998


[ Download a postscript version, MML identifier index, Mizar home page]