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quasar987

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## Main Question or Discussion Point

A topological space X may be defined as

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.

**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.