Journal of Formalized Mathematics
Volume 4, 1992
University of Bialystok
Copyright (c) 1992 Association of Mizar Users

## The Jordan's Property for Certain Subsets of the Plane

Yatsuka Nakamura
Shinshu University, Nagano
Jaroslaw Kotowicz
Warsaw University, Bialystok
The article was written during my visit at Shinshu University in 1992.

### Summary.

Let $S$ be a subset of the topological Euclidean plane ${\cal E}^2_{\rm T}$. We say that $S$ has Jordan's property if there exist two non-empty, disjoint and connected subsets $G_1$ and $G_2$ of ${\cal E}^2_{\rm T}$ such that $S \mathclose{^{\rm c}} = G_1 \cup G_2$ and $\overline{G_1} \setminus G_1 = \overline{G_2} \setminus{G_2}$ (see [13], [8]). The aim is to prove that the boundaries of some special polygons in ${\cal E}^2_{\rm T}$ have this property (see Section 3). Moreover, it is proved that both the interior and the exterior of the boundary of any rectangle in ${\cal E}^2_{\rm T}$ is open and connected.

#### MML Identifier: JORDAN1

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

#### Contents (PDF format)

1. Selected theorems on connected spaces
2. Certain connected and open subsets in the Euclidean plane
3. Jordan's property

#### 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] Agata Darmochwal. The Euclidean space. Journal of Formalized Mathematics, 3, 1991.
[7] Agata Darmochwal and Yatsuka Nakamura. The topological space $\calE^2_\rmT$. Arcs, line segments and special polygonal arcs. Journal of Formalized Mathematics, 3, 1991.
[8] Dick Wick Hall and Guilford L.Spencer II. \em Elementary Topology. John Wiley and Sons Inc., 1955.
[9] Krzysztof Hryniewiecki. Basic properties of real numbers. Journal of Formalized Mathematics, 1, 1989.
[10] Stanislawa Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Journal of Formalized Mathematics, 2, 1990.
[11] Beata Padlewska. Connected spaces. Journal of Formalized Mathematics, 1, 1989.
[12] Beata Padlewska and Agata Darmochwal. Topological spaces and continuous functions. Journal of Formalized Mathematics, 1, 1989.
[13] 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.
[14] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[15] Andrzej Trybulec. A Borsuk theorem on homotopy types. Journal of Formalized Mathematics, 3, 1991.
[16] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[17] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.