Let $(X,\tau)$ be a topological space such that
$\tau\ne\{\emptyset\ X\}.\ $
We call an open cover $\mathcal{U}$ of $(X,\tau)$ *proper* if
$\ X\notin \mathcal{U}.\ $ Moreover we say that $(X,\tau)$ is

*anti-compact*if it does not have a finite proper cover;*anti-paracompact*if for every proper cover $\mathcal{U}$ there is $x\in X$ such that every neighborhood intersects infinitely many members of $\mathcal{U}$;*anti-metacompact*if for every proper cover $\mathcal{U}$ there is $x\in X$ such that $x$ is a member of infinitely many members of $\mathcal{U}$.

We have anti-metacompact $\Rightarrow$ anti-paracompact $\Rightarrow$ anti-compact.

Do any of the converse implications hold?

Illinois J. Math.23 (1979), 241-252. $\endgroup$ – bof Dec 22 '14 at 21:42