Volume 14, 2002

University of Bialystok

Copyright (c) 2002 Association of Mizar Users

**Andrzej Trybulec**- University of Bialystok

- The concept of the minimal distance between two sets in a Euclidean space is introduced and some useful lemmas are proved.

This work has been partially supported by the European Community TYPES grant IST-1999-29001 and CALCULEMUS grant HPRN-CT-2000-00102. The work was completed while the author visited Shinhsu University (Nagano).

- Preliminaries
- Topological and Metrizable Spaces
- Euclid Topological Spaces
- Euclid Plane
- Affine Maps
- Minimal Distance Between Subsets
- BDD and UBD
- Main Definitions

- [1]
Grzegorz Bancerek.
The ordinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [2]
Grzegorz Bancerek and Krzysztof Hryniewiecki.
Segments of natural numbers and finite sequences.
*Journal of Formalized Mathematics*, 1, 1989. - [3]
Jozef Bialas and Yatsuka Nakamura.
The theorem of Weierstrass.
*Journal of Formalized Mathematics*, 7, 1995. - [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.
Partial functions.
*Journal of Formalized Mathematics*, 1, 1989. - [8]
Czeslaw Bylinski and Piotr Rudnicki.
Bounding boxes for compact sets in $\calE^2$.
*Journal of Formalized Mathematics*, 9, 1997. - [9]
Czeslaw Bylinski and Mariusz Zynel.
Cages - the external approximation of Jordan's curve.
*Journal of Formalized Mathematics*, 11, 1999. - [10]
Agata Darmochwal.
Compact spaces.
*Journal of Formalized Mathematics*, 1, 1989. - [11]
Agata Darmochwal.
The Euclidean space.
*Journal of Formalized Mathematics*, 3, 1991. - [12]
Agata Darmochwal and Yatsuka Nakamura.
The topological space $\calE^2_\rmT$. Arcs, line segments and special polygonal arcs.
*Journal of Formalized Mathematics*, 3, 1991. - [13]
Agata Darmochwal and Yatsuka Nakamura.
The topological space $\calE^2_\rmT$. Simple closed curves.
*Journal of Formalized Mathematics*, 3, 1991. - [14]
Adam Grabowski and Yatsuka Nakamura.
The ordering of points on a curve. Part II.
*Journal of Formalized Mathematics*, 9, 1997. - [15]
Krzysztof Hryniewiecki.
Basic properties of real numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [16]
Katarzyna Jankowska.
Matrices. Abelian group of matrices.
*Journal of Formalized Mathematics*, 3, 1991. - [17]
Stanislawa Kanas, Adam Lecko, and Mariusz Startek.
Metric spaces.
*Journal of Formalized Mathematics*, 2, 1990. - [18]
Artur Kornilowicz.
Properties of left and right components.
*Journal of Formalized Mathematics*, 11, 1999. - [19]
Jaroslaw Kotowicz.
Convergent real sequences. Upper and lower bound of sets of real numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [20]
Yatsuka Nakamura.
On Outside Fashoda Meet Theorem.
*Journal of Formalized Mathematics*, 13, 2001. - [21]
Yatsuka Nakamura and Andrzej Trybulec.
Decomposing a Go-Board into cells.
*Journal of Formalized Mathematics*, 7, 1995. - [22]
Yatsuka Nakamura and Andrzej Trybulec.
A decomposition of simple closed curves and the order of their points.
*Journal of Formalized Mathematics*, 9, 1997. - [23]
Yatsuka Nakamura, Andrzej Trybulec, and Czeslaw Bylinski.
Bounded domains and unbounded domains.
*Journal of Formalized Mathematics*, 11, 1999. - [24]
Beata Padlewska.
Connected spaces.
*Journal of Formalized Mathematics*, 1, 1989. - [25]
Beata Padlewska and Agata Darmochwal.
Topological spaces and continuous functions.
*Journal of Formalized Mathematics*, 1, 1989. - [26]
Jan Popiolek.
Some properties of functions modul and signum.
*Journal of Formalized Mathematics*, 1, 1989. - [27]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [28]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [29]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [30]
Edmund Woronowicz.
Relations defined on sets.
*Journal of Formalized Mathematics*, 1, 1989.

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