Jordan Curve Theorem

For simplicity, we adopt the following rules: a, b, c, d, r, s denote real numbers, n denotes a natural number, p, p1, p2 denote points of E 2 T, x, y denote points of E T, C denotes a simple closed curve, A, B, P denote subsets of E 2 T, U , V denote subsets of (E2 T)↾C c, and D denotes a compact middle-intersecting subset of E2 T. Let M be a symmetric triangle Reflexive metric structure and let x, y be points of M . One can verify that ρ(x, y) is non negative. Let n be a natural number and let x, y be points of E T. Note that ρ(x, y) is non negative. Let n be a natural number and let x, y be points of E T. Observe that |x−y| is non negative. We now state several propositions: (1) For all points p1, p2 of E n T such that p1 6= p2 holds 1 2 · (p1 + p2) 6= p1.

[1]  R. Wilder.  On locally connected spaces , 1935 .

[2]  Piotr Rudnicki,et al.  Abian's Fixed Point Theorem , 2007 .

[3]  Agata Darmochwa.  The Topological Space E 2 T . Simple Closed Curves , 2007 .

[4]  Y. Shidama,et al.  Brouwer Fixed Point Theorem for Disks on the Plane , 2005 .

[5]  G. Bancerek,et al.  Homeomorphisms of Jordan Curves , 2005 .

[6]  Adam Grabowski,et al.  On Some Points of a Simple Closed Curve. Part II , 2005 .

[7]  Yasunari Shidama,et al.  Some Properties of Circles on the Plane 1 , 2005 .

[8]  Adam Grabowski,et al.  Algebraic Properties of Homotopies , 2004 .

[9]  Czeslaw Bylinski.  Functions from a Set to a Set , 2004 .

[10]  Czeslaw Bylinski.  Some Basic Properties of Sets , 2004 .

[11]  Krzysztof Hryniewiecki,et al.  Basic Properties of Real Numbers , 2004 .

[12]  Czeslaw Bylinski.  Functions and Their Basic Properties , 2004 .

[13]  Yatsuka Nakamura.  A Decomposition of Simple Closed Curves and the Order of Their Points , 2004 .

[14]  Artur Korniłowicz,et al.  Intersections of Intervals and Balls inE n T , 2004 .

[15]  Yatsuka Nakamura,et al.  The Topological Space E 2 T . Arcs , Line Segments and Special Polygonal Arcs , 2004 .

[16]  Adam Grabowski.  The Ordering of Points on a Curve . Part I , 2004 .

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

[18]  Yatsuka Nakamura,et al.  General Fashoda Meet Theorem for Unit Circle and Square , 2003 .

[19]  Artur Korniłowicz.  The Definition and Basic Properties of Topological Groups , 2002 .

[20]  F. Morel.  An Introduction to a -homotopy Theory , 2002 .

[21]  Czeslaw Bylinski,et al.  Bounding Boxes for Compact Sets in E 2 , 1997 .

[22]  Yatsuka Nakamura,et al.  Extremal Properties of Vertices on Special Polygons, Part I , 1996 .

[23]  Yatsuka Nakamura,et al.  Components and Unions of Components , 1996 .

[24]  Yatsuka Nakamura,et al.  The Jordan's Property for Certain Subsets of the Plane , 1992 .

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

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

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

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

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

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

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

[32]  Edmund Woronowicz.  Relations Defined on Sets , 1990 .

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

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

[35]  Czeslaw Bylinski,et al.  Basic Functions and Operations on Functions , 1989 .

[36]  Czesław Bylí.  Partial functions , 1989 .

[37]  E. Abo-Zeid.  On σ-connected spaces , 1978 .