By Herbert Seifert, William Threlfall
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Extra info for A Textbook of Topology; Topology of 3-Dimensional Fibered Spaces
If S is connected, then the mapping HO is unique as a consequence of the following theorem. 5 (, Chap. 2, Theorem 2). Let X Â IRn , Y Â IRm , and Z Â O HO W Z ! X be two continuous IRk be sets, f W X ! Y be a covering map, and let G; O O O mappings with f ı G D f ı H . z/ for some point O O z 2 Z, then G D H . To formulate the main theorem, we need some additional notions. A set X Â IRn is called path connected if for any two points x0 ; x1 2 X there exists a continuous function p W Œ0; 1 !
Let f :IRn ! Ai ; b i /; i D 1; : : : ; k: If k D 1, then the claim is trivial. x/ D Ai x C b i . x/ D ˛Ak x C ˛b k ¤ Ak ˛x C b k for every positive number ˛ ¤ 1. x/ D Ai xCb i . Ai ; b i /; i D 1; : : : ; k 1, form a collection of matrix–vector pair corresponding to f and thus an induction argument completes the proof that a positively homogeneous piecewise affine function is piecewise linear. It is a matter of plane geometry to check that a real-valued piecewise linear function W IR ! IR of a single variable is positively homogeneous.
24) is equivalent to the identity Ax D x for every x 2 . 1 fx0 g/ \ . 2 fx0 g/ and that the polyhedra A1 . 1 / C b 1 and A2 . n 1/-face if and only if this is true for the polyhedra A. 1 fx0 g/ and 2 fx0 g. A/ > 0. 25) do not hold, then we may replace 1 ; 1 2 fx0 g; 2 fx0 g, A, and I; respectively. n 1/-dimensional linear subspace L Â IRn . Hence there exists an orthogonal matrix Q such that QL D fx 2 IRn jxn D 0g. n 1/-face if and only if this property holds for the polyhedra Q 1 and Q 2 , and A1 x D x holds for every x 2 1 \ 2 if and only if QA1 QT y D y holds for every y 2 Q 1 \ Q 2 .
A Textbook of Topology; Topology of 3-Dimensional Fibered Spaces by Herbert Seifert, William Threlfall