Journal of Formalized Mathematics
Volume 9, 1997
University of Bialystok
Copyright (c) 1997 Association of Mizar Users

Lebesgue's Covering Lemma, Uniform Continuity and Segmentation of Arcs


Yatsuka Nakamura
Shinshu University, Nagano
Andrzej Trybulec
University of Bialystok

Summary.

For mappings from a metric space to a metric space, a notion of uniform continuity is defined. If we introduce natural topologies to the metric spaces, a uniformly continuous function becomes continuous. On the other hand, if the domain is compact, a continuous function is uniformly continuous. For this proof, Lebesgue's covering lemma is also proved. An arc, which is homeomorphic to [0,1], can be divided into small segments, as small as one wishes.

MML Identifier: UNIFORM1

The terminology and notation used in this paper have been introduced in the following articles [20] [24] [21] [16] [13] [1] [2] [22] [19] [18] [25] [3] [5] [6] [23] [11] [17] [8] [7] [10] [9] [12] [14] [4] [15]

Contents (PDF format)

  1. Lebesgue's Covering Lemma
  2. Uniformity of Continuous Functions on Compact Spaces
  3. Segmentation of Arcs

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] Alicia de la Cruz. Totally bounded metric spaces. Journal of Formalized Mathematics, 3, 1991.
[13] Krzysztof Hryniewiecki. Basic properties of real numbers. Journal of Formalized Mathematics, 1, 1989.
[14] Stanislawa Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Journal of Formalized Mathematics, 2, 1990.
[15] Jaroslaw Kotowicz and Yatsuka Nakamura. Introduction to Go-Board --- part I. Journal of Formalized Mathematics, 4, 1992.
[16] Beata Padlewska. Families of sets. Journal of Formalized Mathematics, 1, 1989.
[17] Beata Padlewska and Agata Darmochwal. Topological spaces and continuous functions. Journal of Formalized Mathematics, 1, 1989.
[18] Jan Popiolek. Some properties of functions modul and signum. Journal of Formalized Mathematics, 1, 1989.
[19] Konrad Raczkowski and Pawel Sadowski. Topological properties of subsets in real numbers. Journal of Formalized Mathematics, 2, 1990.
[20] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[21] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.
[22] Michal J. Trybulec. Integers. Journal of Formalized Mathematics, 2, 1990.
[23] Wojciech A. Trybulec. Pigeon hole principle. Journal of Formalized Mathematics, 2, 1990.
[24] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[25] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.

Received November 13, 1997

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