The Theorem of Weierstrass

Jozef Bialas

Lodz \ University, Lodz

Yatsuka Nakamura

Shinshu University, Nagano

Summary.

The basic purpose of this article is to prove the important Weierstrass' theorem which states that a real valued continuous function $f$ on a topological space $T$ assumes a maximum and a minimum value on the compact subset $S$ of $T$, i.e., there exist points $x_1$, $x_2$ of $T$ being elements of $S$, such that $f(x_{1})$ and $f(x_{2})$ are the supremum and the infimum, respectively, of $f(S)$, which is the image of $S$ under the function $f$. The paper is divided into three parts. In the first part, we prove some auxiliary theorems concerning properties of balls in metric spaces and define special families of subsets of topological spaces. These concepts are used in the next part of the paper which contains the essential part of the article, namely the formalization of the proof of Weierstrass' theorem. Here, we also prove a theorem concerning the compactness of images of compact sets of $T$ under a continuous function. The final part of this work is developed for the purpose of defining some measures of the distance between compact subsets of topological metric spaces. Some simple theorems about these measures are also proved.

MML Identifier: WEIERSTR

Contents (PDF format)

1. Preliminaries
2. The Weierstrass' Theorem
3. The Measure of the Distance Between Compact Sets

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 and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Journal of Formalized Mathematics, 1, 1989.

[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] Agata Darmochwal. Compact spaces. Journal of Formalized Mathematics, 1, 1989.

[8] Agata Darmochwal. Families of subsets, subspaces and mappings in topological spaces. Journal of Formalized Mathematics, 1, 1989.

[9] Agata Darmochwal. Finite sets. Journal of Formalized Mathematics, 1, 1989.

[10] Agata Darmochwal. The Euclidean space. Journal of Formalized Mathematics, 3, 1991.

[11] Agata Darmochwal and Yatsuka Nakamura. Metric spaces as topological spaces --- fundamental concepts. Journal of Formalized Mathematics, 3, 1991.

[12] Krzysztof Hryniewiecki. Basic properties of real numbers. Journal of Formalized Mathematics, 1, 1989.

[13] Stanislawa Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Journal of Formalized Mathematics, 2, 1990.

[14] Jaroslaw Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Journal of Formalized Mathematics, 1, 1989.

[15] Beata Padlewska and Agata Darmochwal. Topological spaces and continuous functions. Journal of Formalized Mathematics, 1, 1989.

[16] Jan Popiolek. Some properties of functions modul and signum. Journal of Formalized Mathematics, 1, 1989.

[17] Konrad Raczkowski and Pawel Sadowski. Topological properties of subsets in real numbers. Journal of Formalized Mathematics, 2, 1990.

[18] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.

[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.