Let $ M $ and $ F $ be smooth manifolds. Is the collection of isomorphism classes of fiber bundles of fiber type $ F $ over $ M $ a set or not and why?
2026-03-25 19:04:38.1774465478
Do the isomorphism classes of fiber bundles constitute a set?
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Each such fibre bundle has cardinality $|M|\times|F|$. So take a set of this size, consider the set of all topologies on it, and for each topology, consider the set of all continuous functions to $M$ which have the structure of fibre bundle with fibre $F$. You get a set of fibre bundles which contain representatives of each isomorphism class of bundle.