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It is sometimes necessary to pass from one grating, G, to another, G , by introducing additional lines. Such a new grating G is called a re nement of the old. (It is convenient to agree that G is a re nement of itself.) A common re nement can be formed for any two gratings G1 and G2 , by taking all the lines of G1 and G2 as cross lines. k To each k-chain, C k on G corresponds to the subdivided k-chain C on G ; k k C is the sum of the k-chains into which the k-chain C are subdivided. (0-chains 0 are unaltered by subdivision: C = C 0 .) A subdivided chain has the same locus k k as its original: |C | = |C |. Theorem 5.9.7. [110, V.4.1]Encoding PDF-417 2d Barcode In .NET FrameworkUsing Barcode printer for Visual Studio .NET Control to generate, create PDF 417 image in Visual Studio .NET applications.k k k k (C1 + C2 ) = C1 + C2 , k ( C k ) = (C )PDF 417 Recognizer In .NET FrameworkUsing Barcode reader for Visual Studio .NET Control to read, scan read, scan image in .NET applications.k Corollary 5.9.1. C is a k-cycle if and only if C k is a k-cycle. Encode Barcode In Visual Studio .NETUsing Barcode creation for Visual Studio .NET Control to generate, create barcode image in Visual Studio .NET applications.Separation Theorems. Let E be an open set of T 1 , and let k be a k-cycle on a grating G de ned in E. We say the cycle k bounds in E, denoted as k 0, if there is, on some re nement G of G, a (k + 1)-chain C k+1 such that |C k+1 | E and C k+1 = k on G . We say k is nonbounding in E if | k | E but k does not bound in E. The notion of bounding can be used to study the connectivity of a subset of T 1 : A simple closed curve (a 1-cycle) bounds if it separates a subset from the rest of the space; if two vertices (a 0-cycle) in G do not bound in E, then E is not connected. By Jordan Curve Theorem [110, V.10.2], a simple closed curve always bounds in a plane (a sphere). The following statement indicates this is not necessarily true in a torus.Bar Code Decoder In .NETUsing Barcode scanner for .NET framework Control to read, scan read, scan image in VS .NET applications.Theorem 5.9.8. Every 1-cycle on a rectangular grating in T 1 is the boundary of either none 2-chain or just two 2-chains. Proof: The proof is by induction on the number of lines drawn across the unit square that represents T 1 . On the grating consisting of a and b alone, the only 1-cycles are (a) the null sets, which bound two 2-chains (the zero chain and 2 ), and (b) the cycles a, b, and a + b, each of which does not bound (i.e, does notPDF417 Generator In Visual C#.NETUsing Barcode creator for VS .NET Control to generate, create PDF 417 image in .NET applications.APPENDIX Create PDF 417 In .NETUsing Barcode generator for ASP.NET Control to generate, create PDF-417 2d barcode image in ASP.NET applications.separate a region from T 1 ) because there is only one rectangle in the grating, which is and = 0. Assume the given grating, G1 , is formed from a grating G0 , for which the theorem holds true, by the addition of a line across the square; assume is parallel to a. Let 1 be the given 1-cycle on G1 . We denote by C 2 the sum of the 2-cells of G1 whose lower edges lie in the line and belong to 1 . The 1-cycle 1 + C 2 therefore contains no edge in . 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