## 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?

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 What goes wrong when trying to improve your proof of the last property? – François G. Dorais♦ Apr 20 2010 at 20:06 Basically everything. :-) There's no obvious way to extend it to get anything smaller. – David R. MacIver Apr 20 2010 at 20:30 If it goes that wrong, then you should be able to prove that your space is not paracompact. :-) – François G. Dorais♦ Apr 21 2010 at 0:31 Well, first off there's no way I can prove my space is not paracompact because it's not a single space, and I know all paracompact spaces qualify. :-) But additionally the problem is that (as mentioned elsewhere) it's a parameterised property. The choice of parameter for the case locally < k isn't usefully shrinkable. – David R. MacIver Apr 21 2010 at 7:40 To elaborate, the proof of shrinkability is the one I think most likely to be extendable. The construction is clearly providing more than I'm using in concluding shrinkability (because I can demonstrate that it also rules out $\omega_1$, which is shrinkable), but I can't get a handle on how much more. – David R. MacIver Apr 21 2010 at 8:26

## 2 Answers

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.

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 Your link to the paper seems to be broken. Do you mean it to be jstor.org/pss/2042368 ? Also, could I impose on you to upload the jstor paper to drop.io or similar? I can't get access to very easily. Caryn Navy's spaces all fail to be collectionwise normal, so they don't serve as a counterexample for the whole collection of properties. (by the way, I don't know if you saw my comment on the other answer, but I uploaded a scan of her thesis: drmaciver.com/docs/… ). – David R. MacIver Apr 20 2010 at 8:56 By the way, I think shrinkability is a reasonable amount stronger than normality. It seems to be about the level of strength of a condition as collectionwise normality (although I think neither implies the other) – David R. MacIver Apr 20 2010 at 8:58 I don't have access either, sorry. A disadvantage of not being in academia anymore... I'll see if more can be found on shrinkability. It seems indeed quite strong. I did get your uploaded Navy paper, thanks. – Henno Brandsma Apr 20 2010 at 16:31 Ah, I beg your pardon. I hadn't realised you also weren't in academia any longer. Never mind, I can swing by the British library on the way home and get access to it there. – David R. MacIver Apr 20 2010 at 16:39 Annoyingly I couldn't manage to get an electronic copy at the library. Initial perusal of the paper isn't very encouraging, but will report back on reading it further. – David R. MacIver Apr 20 2010 at 18:45
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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. $\mathbb{R}^{\aleph_0}$?

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 Compact and lindelof aren't options. The properties I'm working with are all implied by paracompactness, so anything stronger than paracompactness isn't possible to prove. Metacompactness / weak paracompactness might be an option - they're equivalent propeties for collectionwise normal spaces - but so far I've not had any luck in proving them. – David R. MacIver Apr 20 2010 at 19:54 Oops. I read "suggestions of avenues to pursue" as an opening for any property... Perhaps you might list the properties you are working with? – Christopher Olah Apr 20 2010 at 20:04 It requires a fair bit of machinery to explain the properties. Essentially it's a parameterized property we're calling OCA(V), where V is a normed space. The property is basically designed to let you construct continuous functions from X into V. – David R. MacIver Apr 20 2010 at 20:23