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?

thatwrong, then you should be able to prove that your space is not paracompact. :-) $\endgroup$isshrinkable), but I can't get a handle on how much more. $\endgroup$