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

Contents (PDF format)

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

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