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Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
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On the continuity of a function given by evaluating compact subsets of smooth functions
Let $M$ be a compact connected smooth manifold. Write $C^{\infty}(M)$ for the Frechet space of the smooth real-valued functions on $M$ equipped with the usual $C^{\infty}$-topology.
Given a compact su …