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Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.

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Approximate selection for finite-valued upper hemicontinuous/semicontinuous maps?

I'd like to know if there are any known-results on the existence of continuous approximation theorems for upper hemicontinuous (aka upper semicontinuous) maps $\phi: X\rightarrow Y$ which are finite v …
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