Volume 4, 1992

University of Bialystok

Copyright (c) 1992 Association of Mizar Users

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

- 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.

- Selected theorems on connected spaces
- Certain connected and open subsets in the Euclidean plane
- Jordan's property

- [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.
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Czeslaw Bylinski.
Functions from a set to a set.
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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.
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Tarski Grothendieck set theory.
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A Borsuk theorem on homotopy types.
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Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989.

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