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**paracompact**by the condition that every open cover U of X admits a refinement U' such that every point of X intersects only a finite number of elements of U'.

A seemingly stronger condition on X would be that every open cover U of X admits a refinement U' such that around every point x of X, there is an open nbhd A which intersects only finitely many elements of U'.

I'm pretty sure that in fact these conditions are equivalent (at least for metric spaces) but I'm having trouble proving it.