Volume 4, 1992

University of Bialystok

Copyright (c) 1992 Association of Mizar Users

**Toshihiko Watanabe**- Shinshu University, Nagano

- 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.

- Properties of Topological Intervals
- Continuous Mappings Between Topological Intervals
- Connectedness of Intervals and Brouwer Fixed Point Theorem for Intervals

- [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.

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