Anyone can give the example? It is an exercise of Boothby's Manifolds book.
2026-04-11 10:55:49.1775904949
Provide an example of a locally Euclidean, paracompact Hausdorff space but not have a countable basis.
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A perfectly good example would be an uncountable space with the discrete topology. If you want something slightly less trivial, take the Cartesian product of this space with any locally Euclidean, paracompact Hausdorff space.