Conditions useful for proving paracompactness I have a family of properties which I want to show taken together imply paracompactness (I can show that they are all implied by paracompactness). I can prove a whole bunch of things which are consequences of paracompactness, and was wondering if given any of these there's some simpler property equivalent to paracompactness I could prove. Any suggestions would be welcome.
Given X with these properties I can prove:


*

*X is normal

*X is countably paracompact

*X is collectionwise normal

*Every open cover $\{ U_a \}$ can be shrunk to a closed cover $\{ F_a \}$ with $F_a \subseteq U_a$. (I assume this property isn't equivalent to paracompactness? I know it's equivalent to countable paracompactness when the set of $U_a$ is countable, and I know if you add "locally finite" to the condition it becomes equivalent to paracompactness)

*Every open cover of X by $\kappa$ many open sets, where $\kappa$ is regular, has an open refinement which is locally $< \kappa$.


I don't think together these are sufficient to prove paracompactness, though I don't have a counter example. I believe $\omega_1$ satisfies all the properties but the last, though I've not confirmed you can shrink open covers to closed (it looks plausible though).
Any suggestions of avenues to pursue?
 A: This paper has some theorems on shrinkable covers, which might be helpful. It's pretty close to normality, it seems (I knew that we can shrink locally finite covers in normal spaces). This book has some connections to $\kappa$-Dowker spaces that might be useful.
It is mentioned that Navy's space is also shrinking (i.e. has your fourth property), so it might be a candidate counterexample. 
A: Well, there's always compact or Lindelof + T3. I actually made a graph about that!

Beyond that... Well, from counterexamples in topology we find out that:
Fully Normal -> Paracompact
T2 + Fully T4 -> Paracompact
As for my personal thoughts: there's probably some way to reverse paracompactness -> metacompactness:
Paracompact: Every open cover locally finite.
Metacompact: Every open cover point finite.
You'd need a pretty odd space to not have them be equivalent. Specifically: there exists some point with which every open cover has finite intersections but any open set containing it has non-finite intersections...
I suppose a non-finite number of sets approach each point from different directions so that they don't overlap too much? If you think about that in terms of dimensions, it's very weird. $\mathbb{R}^{\aleph_0}$? 
