The Jordan's Property for Certain Subsets of the Plane

Let S be a subset of the topological Euclidean plane E T. We say that S has Jordan’s property if there exist two non-empty, disjoint and connected subsets G1 and G2 of E 2 T such that S c = G1 ∪ G2 and G1 \ G1 = G2 \ G2 (see [19], [10]). The aim is to prove that the boundaries of some special polygons in E 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 E T is open and connected.

[1]  A. Trybulec.  Domains and Their Cartesian Products , 1990 .

[2]  Beata Padlewska,et al.  Families of Sets , 1990 .

[3]  Dariusz Surowik,et al.  Cyclic Groups and Some of Their Properties - Part I , 1991 .

[4]  Konrad Raczkowski,et al.  Topological Properties of Subsets in Real Numbers , 1990 .

[5]  Andrzej Trybulec,et al.  Tuples, Projections and Cartesian Products , 1990 .

[6]  Zbigniew Karno,et al.  The Lattice of Domains of an Extremally Disconnected Space 1 , 1992 .

[7]  G. Bancerek,et al.  Ordinal Numbers , 2003 .

[8]  lawa Kanas,et al.  Metric Spaces , 2020, An Introduction to Functional Analysis.

[9]  Wojciech A. Trybulec.  Subgroup and Cosets of Subgroups , 1990 .

[10]  Grzegorz Bancerek,et al.  Segments of Natural Numbers and Finite Sequences , 1990 .

[11]  Zbigniew Karno,et al.  Separated and Weakly Separated Subspaces of Topological Spaces , 1991 .

[12]  Yatsuka Nakamura,et al.  Metric Spaces as Topological Spaces - Fundamental Concepts , 1991 .

[13]  Wojciech A. Trybulec.  Vectors in Real Linear Space , 1990 .

[14]  Katarzyna Jankowska,et al.  Transpose Matrices and Groups of Permutations , 1991 .

[15]  Alicia de la Cruz.  Introduction to Modal Propositional Logic , 1991 .

[16]  Micha l Muzalewski,et al.  Three-Argument Operations and Four-Argument Operations 1 , 1991 .

[17]  Wojciech A. Trybulec.  Linear Combinations in Vector Space , 1990 .

[18]  Beata Padlewska.  Connected Spaces , 1989 .

[19]  B. Balkay,et al.  Introduction to lattice theory , 1965 .

[20]  Micha l Muzalewski,et al.  Operations on Submodules in Left Module over Associative Ring 1 , 1991 .

[21]  Wojciech A. Trybulec.  Subspaces and Cosets of Subspaces in Real Linear Space , 1990 .

[22]  K. Prazmowski,et al.  A Construction of Analytical Projective Space , 1990 .

[23]  A. Trybulec.  Tarski Grothendieck Set Theory , 1990 .

[24]  Edwin E. Moise.  The Jordan curve theorem , 1977 .

[25]  Czesław Bylí,et al.  Binary Operations , 2019, Problem Solving in Mathematics and Beyond.

[26]  Andrzej Trybulec,et al.  Binary Operations Applied to Functions , 1990 .

[27]  Grzegorz Bancerek,et al.  Cartesian Product of Functions , 1991 .

[28]  Wojciech A. Trybulec.  Operations on Subspaces in Vector Space , 1990 .

[29]  Zbigniew Karno.  Remarks on Special Subsets of Topological Spaces , 1992 .

[30]  Micha l Muzalewski.  Rings and Modules - Part II , 1991 .

[31]  Andrzej Trybulec,et al.  A Borsuk Theorem on Homotopy Types , 1991 .

[32]  Wojciech A. Trybulec,et al.  Homomorphisms and Isomorphisms of Groups. Quotient Group , 1991 .

[33]  J. Kieran,et al.  Introduction to Trees , 1960 .

[34]  Zbigniew Karno,et al.  Continuity of Mappings over the Union of Subspaces , 1992 .

[35]  Konrad Raczkowski,et al.  Topological Properties of Subsets in Real Numbers 1 , 1990 .

[36]  Wojciech A. Trybulec.  Basis of Real Linear Space , 1990 .

[37]  Wojciech A. Trybulec.  Linear Combinations in Real Linear Space , 1990 .

[38]  Agata Darmochwa,et al.  Topological Spaces and Continuous Functions , 1990 .

[39]  Yatsuka Nakamura,et al.  A Mathematical Model of CPU , 1992 .

[40]  Wojciech A. Trybulec.  Binary Operations on Finite Sequences , 1990 .

[41]  Wojciech A. Trybulec.  Subspaces and Cosets of Subspaces in Vector Space , 1990 .

[42]  Jaros law Kotowicz,et al.  Convergent Real Sequences . Upper and Lower Bound of Sets of Real Numbers , 1989 .

[43]  Agata Darmochwał.  Families of Subsets , Subspaces and Mappings in Topological Spaces , 1989 .

[44]  Agata Darmochwa.  Euclidean Space , 2018, How to Pass the FRACP Written Examination.

[45]  Ben Dushnik,et al.  Partially Ordered Sets , 1941 .

[46]  G. Bancerek.  The Fundamental Properties of Natural Numbers , 1990 .

[47]  Wojciech A. Trybulec.  Pigeon Hole Principle , 1990 .

[48]  Leszek Borys,et al.  Paracompact and Metrizable Spaces , 1991 .

[49]  Micha l Muzalewski,et al.  Submodules and Cosets of Submodules in Left Module over Associative Ring 1 , 1991 .

[50]  Wojciech A. Trybulec.  BASIS FOR A VECTOR SPACE , 1990 .