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

## The Ordering of Points on a Curve. Part II

University of Bialystok
This paper was written while the author visited the Shinshu University in the winter of 1997.
Yatsuka Nakamura
Shinshu University, Nagano

### Summary.

The proof of the Jordan Curve Theorem according to [11] is continued. The notions of the first and last point of a oriented arc are introduced as well as ordering of points on a curve in \$\calE^2_T\$.

#### MML Identifier: JORDAN5C

The terminology and notation used in this paper have been introduced in the following articles [12] [14] [1] [15] [2] [3] [4] [7] [13] [9] [8] [10] [5] [6]

#### Contents (PDF format)

1. First and last point of a curve
2. The ordering of points on a curve
3. Some properties of the ordering of points on a curve

#### Bibliography

[1] Grzegorz Bancerek. The fundamental properties of natural 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] 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] Agata Darmochwal. Families of subsets, subspaces and mappings in topological spaces. Journal of Formalized Mathematics, 1, 1989.
[6] Agata Darmochwal. The Euclidean space. Journal of Formalized Mathematics, 3, 1991.
[7] Agata Darmochwal and Yatsuka Nakamura. Metric spaces as topological spaces --- fundamental concepts. Journal of Formalized Mathematics, 3, 1991.
[8] Agata Darmochwal and Yatsuka Nakamura. The topological space \$\calE^2_\rmT\$. Arcs, line segments and special polygonal arcs. Journal of Formalized Mathematics, 3, 1991.
[9] Yatsuka Nakamura and Roman Matuszewski. Reconstructions of special sequences. Journal of Formalized Mathematics, 8, 1996.
[10] Beata Padlewska and Agata Darmochwal. Topological spaces and continuous functions. Journal of Formalized Mathematics, 1, 1989.
[11] Yukio Takeuchi and Yatsuka Nakamura. On the Jordan curve theorem. Technical Report 19804, Dept. of Information Eng., Shinshu University, 500 Wakasato, Nagano city, Japan, April 1980.
[12] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[13] Wojciech A. Trybulec. Pigeon hole principle. Journal of Formalized Mathematics, 2, 1990.
[14] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[15] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.