Journal of Formalized Mathematics
Volume 11, 1999
University of Bialystok
Copyright (c) 1999 Association of Mizar Users

## Bounded Domains and Unbounded Domains

Yatsuka Nakamura
Shinshu University, Nagano
Andrzej Trybulec
University of Bialystok
Czeslaw Bylinski
University of Bialystok

### Summary.

First, notions of inside components and outside components are introduced for any subset of \$n\$-dimensional Euclid space. Next, notions of the bounded domain and the unbounded domain are defined using the above components. If the dimension is larger than 1, and if a subset is bounded, a unbounded domain of the subset coincides with an outside component (which is unique) of the subset. For a sphere in \$n\$-dimensional space, the similar fact is true for a bounded domain. In 2 dimensional space, any rectangle also has such property. We discussed relations between the Jordan property and the concept of boundary, which are necessary to find points in domains near a curve. In the last part, we gave the sufficient criterion for belonging to the left component of some clockwise oriented finite sequences.

#### MML Identifier: JORDAN2C

The terminology and notation used in this paper have been introduced in the following articles [38] [9] [45] [32] [46] [7] [8] [3] [40] [18] [2] [1] [34] [47] [13] [20] [6] [31] [33] [17] [29] [36] [15] [4] [10] [44] [41] [35] [5] [21] [30] [37] [24] [11] [14] [26] [12] [43] [42] [16] [19] [22] [27] [23] [28] [39] [25]

#### Contents (PDF format)

1. Definitions of Bounded Domain and Unbounded Domain
2. Bounded and Unbounded Domains of Rectangles
3. Jordan Property and Boundary Property
4. Points in LeftComp

#### Bibliography

[1] Grzegorz Bancerek. Cardinal numbers. Journal of Formalized Mathematics, 1, 1989.
[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Journal of Formalized Mathematics, 1, 1989.
[3] Grzegorz Bancerek. The ordinal numbers. Journal of Formalized Mathematics, 1, 1989.
[4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Journal of Formalized Mathematics, 1, 1989.
[5] Jozef Bialas and Yatsuka Nakamura. The theorem of Weierstrass. Journal of Formalized Mathematics, 7, 1995.
[6] Leszek Borys. Paracompact and metrizable spaces. Journal of Formalized Mathematics, 3, 1991.
[7] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[8] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[9] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.
[10] Czeslaw Bylinski. Finite sequences and tuples of elements of a non-empty sets. Journal of Formalized Mathematics, 2, 1990.
[11] Czeslaw Bylinski. The sum and product of finite sequences of real numbers. Journal of Formalized Mathematics, 2, 1990.
[12] Czeslaw Bylinski and Piotr Rudnicki. Bounding boxes for compact sets in \$\calE^2\$. Journal of Formalized Mathematics, 9, 1997.
[13] Agata Darmochwal. Compact spaces. Journal of Formalized Mathematics, 1, 1989.
[14] Agata Darmochwal. The Euclidean space. Journal of Formalized Mathematics, 3, 1991.
[15] Agata Darmochwal and Yatsuka Nakamura. Metric spaces as topological spaces --- fundamental concepts. Journal of Formalized Mathematics, 3, 1991.
[16] Agata Darmochwal and Yatsuka Nakamura. The topological space \$\calE^2_\rmT\$. Arcs, line segments and special polygonal arcs. Journal of Formalized Mathematics, 3, 1991.
[17] Alicia de la Cruz. Totally bounded metric spaces. Journal of Formalized Mathematics, 3, 1991.
[18] Krzysztof Hryniewiecki. Basic properties of real numbers. Journal of Formalized Mathematics, 1, 1989.
[19] Katarzyna Jankowska. Matrices. Abelian group of matrices. Journal of Formalized Mathematics, 3, 1991.
[20] Stanislawa Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Journal of Formalized Mathematics, 2, 1990.
[21] Jaroslaw Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Journal of Formalized Mathematics, 1, 1989.
[22] Jaroslaw Kotowicz and Yatsuka Nakamura. Introduction to Go-Board --- part I. Journal of Formalized Mathematics, 4, 1992.
[23] Jaroslaw Kotowicz and Yatsuka Nakamura. Introduction to Go-Board --- part II. Journal of Formalized Mathematics, 4, 1992.
[24] Eugeniusz Kusak, Wojciech Leonczuk, and Michal Muzalewski. Abelian groups, fields and vector spaces. Journal of Formalized Mathematics, 1, 1989.
[25] Roman Matuszewski and Yatsuka Nakamura. Projections in \$n\$-dimensional Euclidean space to each coordinates. Journal of Formalized Mathematics, 9, 1997.
[26] Yatsuka Nakamura and Czeslaw Bylinski. Extremal properties of vertices on special polygons, part I. Journal of Formalized Mathematics, 6, 1994.
[27] Yatsuka Nakamura and Jaroslaw Kotowicz. The Jordan's property for certain subsets of the plane. Journal of Formalized Mathematics, 4, 1992.
[28] Yatsuka Nakamura and Andrzej Trybulec. Decomposing a Go-Board into cells. Journal of Formalized Mathematics, 7, 1995.
[29] Yatsuka Nakamura and Andrzej Trybulec. Components and unions of components. Journal of Formalized Mathematics, 8, 1996.
[30] Takaya Nishiyama and Yasuho Mizuhara. Binary arithmetics. Journal of Formalized Mathematics, 5, 1993.
[31] Beata Padlewska. Connected spaces. Journal of Formalized Mathematics, 1, 1989.
[32] Beata Padlewska. Families of sets. Journal of Formalized Mathematics, 1, 1989.
[33] Beata Padlewska. Locally connected spaces. Journal of Formalized Mathematics, 2, 1990.
[34] Beata Padlewska and Agata Darmochwal. Topological spaces and continuous functions. Journal of Formalized Mathematics, 1, 1989.
[35] Jan Popiolek. Some properties of functions modul and signum. Journal of Formalized Mathematics, 1, 1989.
[36] Agnieszka Sakowicz, Jaroslaw Gryko, and Adam Grabowski. Sequences in \$\calE^N_\rmT\$. Journal of Formalized Mathematics, 6, 1994.
[37] Andrzej Trybulec. Binary operations applied to functions. Journal of Formalized Mathematics, 1, 1989.
[38] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[39] Andrzej Trybulec. Left and right component of the complement of a special closed curve. Journal of Formalized Mathematics, 7, 1995.
[40] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.
[41] Andrzej Trybulec and Czeslaw Bylinski. Some properties of real numbers operations: min, max, square, and square root. Journal of Formalized Mathematics, 1, 1989.
[42] Andrzej Trybulec and Yatsuka Nakamura. On the order on a special polygon. Journal of Formalized Mathematics, 9, 1997.
[43] Andrzej Trybulec and Yatsuka Nakamura. On the rectangular finite sequences of the points of the plane. Journal of Formalized Mathematics, 9, 1997.
[44] Wojciech A. Trybulec. Pigeon hole principle. Journal of Formalized Mathematics, 2, 1990.
[45] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[46] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[47] Miroslaw Wysocki and Agata Darmochwal. Subsets of topological spaces. Journal of Formalized Mathematics, 1, 1989.

Received January 7, 1999