General Fashoda Meet Theorem for Unit Circle

Yatsuka Nakamura

Shinshu University, Nagano

Summary.

Outside and inside Fashoda theorems are proven for points in general position on unit circle. Four points must be ordered in a sense of ordering for simple closed curve. For preparation of proof, the relation between the order and condition of coordinates of points on unit circle is discussed.

MML Identifier: JGRAPH_5

Contents (PDF format)

1. Preliminaries
2. Fashoda Meet Theorems for Circle in Special Case
3. Properties of Fan Morphisms
4. Order of Points on Circle
5. General Fashoda Theorems

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. Binary operations. Journal of Formalized Mathematics, 1, 1989.

[4] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.

[5] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.

[6] Czeslaw Bylinski and Piotr Rudnicki. Bounding boxes for compact sets in $\calE^2$. Journal of Formalized Mathematics, 9, 1997.

[7] Agata Darmochwal. Compact spaces. Journal of Formalized Mathematics, 1, 1989.

[8] Agata Darmochwal. Families of subsets, subspaces and mappings in topological spaces. Journal of Formalized Mathematics, 1, 1989.

[9] Agata Darmochwal. The Euclidean space. Journal of Formalized Mathematics, 3, 1991.

[10] Agata Darmochwal and Yatsuka Nakamura. Metric spaces as topological spaces --- fundamental concepts. Journal of Formalized Mathematics, 3, 1991.

[11] Krzysztof Hryniewiecki. Basic properties of real numbers. Journal of Formalized Mathematics, 1, 1989.

[12] Stanislawa Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Journal of Formalized Mathematics, 2, 1990.

[13] Yatsuka Nakamura. Fan homeomorphisms in the plane. Journal of Formalized Mathematics, 14, 2002.

[14] Yatsuka Nakamura and Andrzej Trybulec. A decomposition of simple closed curves and the order of their points. Journal of Formalized Mathematics, 9, 1997.

[15] Beata Padlewska and Agata Darmochwal. Topological spaces and continuous functions. Journal of Formalized Mathematics, 1, 1989.

[16] Konrad Raczkowski and Pawel Sadowski. Topological properties of subsets in real numbers. Journal of Formalized Mathematics, 2, 1990.

[17] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.

[18] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.

[19] Andrzej Trybulec and Czeslaw Bylinski. Some properties of real numbers operations: min, max, square, and square root. Journal of Formalized Mathematics, 1, 1989.

[20] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.