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A generalization of Steenrod’s approximation theorem. (English) Zbl 1212.58005
Summary: In this paper we aim for a generalization of the Steenrod approximation theorem from Section 6.7 in [J. Steenrod, The topology of fibre bundles. Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press (1999; Zbl 0942.55002)] concerning a smoothing procedure for sections in smooth locally trivial bundles. The generalization is that we consider locally trivial smooth bundles with a possibly infinite-dimensional typical fibre. The main result states that a continuous section in a smooth locally trivial bundles can always be smoothed out in a very controlled way (in terms of the graph topology on spaces of continuous functions) preserving the section on regions where it is already smooth.

MSC:
58B05 Homotopy and topological questions for infinite-dimensional manifolds
57R10 Smoothing in differential topology
57R12 Smooth approximations in differential topology
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