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Consider a topological space $X$. What can be said about the following property?


For any open cover $\mathcal U = \{ U_i \}_{ i \in I }$ of $X$,
there exists an open refinement $\mathcal V = \{ V_j \}_{ j \in J }$ such that any set $V \in \mathcal V$ of the refinement intersects only finitely many other sets of $\mathcal V$.


This property is stronger than paracompactness, but is it weaker than compactness?

Does it hold on topological manifolds (which are paracompact and second-countable)?
