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10 Local classification theorems
12 Pathwise connectedness around a point
13 Local homotopy groups
5 other sections not shown
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admissible extension admissible homotopy admissible map basic path coefficient group cohomology compact-open topology completely regular completes the proof complex numbers conic neighborhood conic point connected around xQ continuous real function Corollary countable basis define a homotopy Define a path Define an admissible deformation retract deg g denote the natural denote the subspace element exists a homotopy exists a path exists an open f and g f induces following property following Theorem given local map given triplet grotp Hausdorff space homo topic homology theory homotopy classes homotopy groups Hopf theorem identity map immediate consequence inclusion map ind f induced homomorphisms integer isomorphisms Let f local homotopy groups locally homotopic map f map of X,x maps f,g n-connected natural projection neighborhood of xQ open neighborhood pathwise accessible pathwise connected point xQ positive real number singular homology sub space tangent space topological space X,xQ