Journal of Formalized Mathematics
Volume 4, 1992
University of Bialystok
Copyright (c) 1992 Association of Mizar Users

The Brouwer Fixed Point Theorem for Intervals


Toshihiko Watanabe
Shinshu University, Nagano

Summary.

The aim is to prove, using Mizar System, the following simplest version of the Brouwer Fixed Point Theorem [3]. {\em For every continuous mapping $f : {\Bbb I} \rightarrow {\Bbb I}$ of the topological unit interval $\Bbb I$ there exists a point $x$ such that $f(x) = x$} (see e.g. [9], [4]).

This paper was done under the supervision of Z. Karno while the author was visiting the Institute of Mathematics of Warsaw University in Bia{\l}ystok.

MML Identifier: TREAL_1

The terminology and notation used in this paper have been introduced in the following articles [17] [20] [1] [19] [21] [5] [6] [10] [16] [15] [7] [14] [11] [13] [2] [8] [12] [18]

Contents (PDF format)

  1. Properties of Topological Intervals
  2. Continuous Mappings Between Topological Intervals
  3. Connectedness of Intervals and Brouwer Fixed Point Theorem for Intervals

Acknowledgments

The author wishes to express his thanks to Professors A.~Trybulec and Z.~Karno for their useful suggestions and many valuable comments.

Bibliography

[1] Grzegorz Bancerek. The ordinal numbers. Journal of Formalized Mathematics, 1, 1989.
[2] Leszek Borys. Paracompact and metrizable spaces. Journal of Formalized Mathematics, 3, 1991.
[3] L. Brouwer. \"Uber Abbildungen von Mannigfaltigkeiten. \em Mathematische Annalen, 38(71):97--115, 1912.
[4] Robert H. Brown. \em The Lefschetz Fixed Point Theorem. Scott--Foresman, New York, 1971.
[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] Agata Darmochwal. Families of subsets, subspaces and mappings in topological spaces. Journal of Formalized Mathematics, 1, 1989.
[8] Agata Darmochwal and Yatsuka Nakamura. Metric spaces as topological spaces --- fundamental concepts. Journal of Formalized Mathematics, 3, 1991.
[9] James Dugundji and Andrzej Granas. \em Fixed Point Theory, volume I. PWN - Polish Scientific Publishers, Warsaw, 1982.
[10] Krzysztof Hryniewiecki. Basic properties of real numbers. Journal of Formalized Mathematics, 1, 1989.
[11] Stanislawa Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Journal of Formalized Mathematics, 2, 1990.
[12] Zbigniew Karno. Separated and weakly separated subspaces of topological spaces. Journal of Formalized Mathematics, 4, 1992.
[13] Michal Muzalewski. Categories of groups. Journal of Formalized Mathematics, 3, 1991.
[14] Beata Padlewska. Connected spaces. Journal of Formalized Mathematics, 1, 1989.
[15] Beata Padlewska and Agata Darmochwal. Topological spaces and continuous functions. Journal of Formalized Mathematics, 1, 1989.
[16] Konrad Raczkowski and Pawel Sadowski. Topological properties of subsets in real numbers. Journal of Formalized Mathematics, 2, 1990.
[17] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[18] Andrzej Trybulec. A Borsuk theorem on homotopy types. Journal of Formalized Mathematics, 3, 1991.
[19] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.
[20] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[21] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.

Received August 17, 1992

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