Let $(X,\tau)$ be a topological space, and let ${\cal U}$ be an open cover. We say that ${\cal U}$ is *thick* if for all $x\in X$ we have $$|\{V\in {\cal U}: x\in V\}| = |X|.$$
Is there a space $(X,\tau)$ and an open cover ${\cal U}$ such that every [refinement](https://en.wikipedia.org/wiki/Cover_(topology)#Refinement) of ${\cal U}$ is thick?