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)?
This property has been called hypocompactness; it is strictly weaker than compactness and strictly stronger than paracompactness. The Sorgenfrey line is hypocompact but not compact: every open cover has a disjoint clopen refinement. Any hedgehog space $X$ of uncountable spininess is an example of a paracompact space that is not hypocompact. $X$ is metrizable, so it’s paracompact. If $X$ has spininess $\kappa$, $p$ is the centre point of the hedgehog, the points at the ends of the spines are $q_\xi$ for $\xi<\kappa$, and the metric $d$ is as in the linked article, the open cover
$$\left\{B_d\left(p,\frac23\right)\right\}\cup\left\{B_d\left(q_\xi,\frac23\right):\xi<\kappa\right\}$$
has no open refinement that is star-finite at $p$.
Added: Henno reminds me that I forgot to answer the last question. Paracompact Lindelöf spaces are hypocompact, so separable metric spaces are hypocompact.