Size Homotopy Group
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High Quality Content by WIKIPEDIA articles! The concept of size homotopy group is analogous in size theory of the classical concept of homotopy group. In order to give its definition, let us assume that a size pair (M,varphi) is given, where M is a closed manifold of class C^0 and varphi:Mto mathbb{R}^k is a continuous function. Let us consider the partial order preceq in mathbb{R}^k defined by setting (x_1,ldots,x_k)preceq(y_1,ldots,y_k) if and only if x_1 le y_1,ldots, x_k le y_k . For every Yinmathbb{R}^k we set M_{Y}={Zinmathbb{R}^k:Zpreceq Y} . Assume that Pin M_X and Xpreceq Y . If alpha...
High Quality Content by WIKIPEDIA articles! The concept of size homotopy group is analogous in size theory of the classical concept of homotopy group. In order to give its definition, let us assume that a size pair (M,varphi) is given, where M is a closed manifold of class C^0 and varphi:Mto mathbb{R}^k is a continuous function. Let us consider the partial order preceq in mathbb{R}^k defined by setting (x_1,ldots,x_k)preceq(y_1,ldots,y_k) if and only if x_1 le y_1,ldots, x_k le y_k . For every Yinmathbb{R}^k we set M_{Y}={Zinmathbb{R}^k:Zpreceq Y} . Assume that Pin M_X and Xpreceq Y . If alpha , beta are two paths from P to P and a homotopy from alpha to beta , based at P , exists in the topological space M_{Y} , then we write alpha approx_{Y}beta . The first size homotopy group of the size pair (M,varphi) computed at (X,Y) is defined to be the quotient set of the set of all paths from P to P in M_X with respect to the equivalence relation approx_{Y} , endowed with the operation induced by the usual composition of based loops.
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