Volume 11, 1999

University of Bialystok

Copyright (c) 1999 Association of Mizar Users

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

- 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.

- Definitions of Bounded Domain and Unbounded Domain
- Bounded and Unbounded Domains of Rectangles
- Jordan Property and Boundary Property
- Points in LeftComp

- [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.

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