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

## A Decomposition of Simple Closed Curves and the Order of Their Points

Yatsuka Nakamura
Shinshu University, Nagano
Andrzej Trybulec
University of Bialystok

### Summary.

The goal of the article is to introduce an order on a simple closed curve. To do this, we fix two points on the curve and devide it into two arcs. We prove that such a decomposition is unique. Other auxiliary theorems about arcs are proven for preparation of the proof of the above.

#### MML Identifier: JORDAN6

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

#### Contents (PDF format)

1. Middle Points of Arcs
2. Segments of Arcs
3. Decomposition of a Simple Closed Curve Into Two Arcs
4. Uniqueness of Decomposition of a Simple Closed Curve
5. Lower Arcs and Upper Arcs
6. An Order of Points in a Simple Closed Curve

#### Bibliography

[1] Grzegorz Bancerek. The ordinal numbers. Journal of Formalized Mathematics, 1, 1989.
[2] Leszek Borys. Paracompact and metrizable spaces. Journal of Formalized Mathematics, 3, 1991.
[3] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[4] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[5] Czeslaw Bylinski and Piotr Rudnicki. Bounding boxes for compact sets in \$\calE^2\$. Journal of Formalized Mathematics, 9, 1997.
[6] Agata Darmochwal. Compact spaces. Journal of Formalized Mathematics, 1, 1989.
[7] Agata Darmochwal. The Euclidean space. Journal of Formalized Mathematics, 3, 1991.
[8] Agata Darmochwal and Yatsuka Nakamura. Metric spaces as topological spaces --- fundamental concepts. Journal of Formalized Mathematics, 3, 1991.
[9] Agata Darmochwal and Yatsuka Nakamura. The topological space \$\calE^2_\rmT\$. Arcs, line segments and special polygonal arcs. Journal of Formalized Mathematics, 3, 1991.
[10] Agata Darmochwal and Yatsuka Nakamura. The topological space \$\calE^2_\rmT\$. Simple closed curves. Journal of Formalized Mathematics, 3, 1991.
[11] Adam Grabowski and Yatsuka Nakamura. The ordering of points on a curve. Part II. Journal of Formalized Mathematics, 9, 1997.
[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. Connected spaces. Journal of Formalized Mathematics, 1, 1989.
[16] Beata Padlewska and Agata Darmochwal. Topological spaces and continuous functions. 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. A Borsuk theorem on homotopy types. Journal of Formalized Mathematics, 3, 1991.
[20] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.
[21] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[22] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.