Are there known examples of compact infinite dimensional manifolds?
The word "manifold" is important.
closed as unclear what you're asking by BS., Andrey Rekalo, Carlo Beenakker, Ricardo Andrade, David White Oct 25 '13 at 12:44
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The empty space is a manifold of any dimension.
No, seriously, let's assume that "manifold" means a Hausdorff space in which every point has an open neighborhood homeomorphic to an open subset of a topological vector space. If the manifold is compact and nonempty then the vector space must be locally compact. As far as I know, that makes it finite-dimensional.