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Is there a nice characterization of topological spaces with the property that the product with any paracompact space is paracompact?

All compact spaces have this property (this can be shown from the tube lemma). But somebody once gave me an example (that I cannot locate) of a non-compact space with the property. I didn't check the example carefully, so I cannot vouch for its accuracy.

If a characterization is too hard, an example of a non-compact space would also be great.

[NOTE: I don't assume Hausdorffness in my definitions of compact and paracompact, though it would be nice if the example were a Hausdorff space.]

ADDED LATER: I forgot to mention this, but a product of paracompact spaces need not be paracompact. The standard example is the Sorgenfrey line (the real line with the lower limit topology), which is paracompact, whose product with itself, the Sorgenfrey plane, is not paracompact.

edited Jun 23 2011 at 13:15

Stephen S
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asked Jan 8 2010 at 15:02

Vipul Naik
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Discrete spaces have this property (and may be non-compact). – a-fortiori Jan 8 2010 at 16:06

I'm confused. Isn't a point paracompact? If (X x Y) is paracompact for any paracompact Y, X = (X x pt) is paracompact? – YBL Jan 29 2010 at 23:14

Yes, that's true, so if a space has this property, it must be paracompact. But not all paracompact spaces have this property, as the Sorgenfrey line example illustrates. I was looking for a deeper characterization. – Vipul Naik Feb 7 2010 at 3:40

François G. Dorais · Accepted Answer · 2010-03-16 04:38:10Z UTC

One interesting conjectured characterization is due to Rastislav Telgársky. In Spaces defined by topological games (Fund. Math. 88, 1975), Telgársky coined several games and provided partial results regarding this class of paracompact spaces (among other things). In Some remarks on a Telgarsky’s conjecture concerning products of paracompact spaces (Topology Appl. 156, 2009, MR2512606), Kazimierz Alster made some significant progress in showing that one of Telgársky's games completely characterizes the class of spaces whose product with every paracompact space is paracompact.

Looking at MathSciNet, it looks like Alster is getting close to answering this question (MR2243730 and MR2502008).

David Carchedi · Answer 2 · 2010-03-25 00:07:12Z UTC

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The product of a CW-complex with a paracompact-Hausdorff space is again paracompact Hausdorff. I'm not sure what happens if you remove the word Hausdorff.

answered Mar 25 2010 at 0:07

David Carchedi
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Thanks! Do you have a reference for the statement/proof? I can then check it up to see how the proof works out. – Vipul Naik Mar 25 2010 at 20:43

In fact, I have no reference. It's just one of those facts that I've "picked up". I've read it in a few papers and even used the fact in one of my own. If you find a reference, I'd be interested. P.S. An interested class of paracompact-spaces closed under finite products is geometric realizations of simplicial sets- SINCE, geometric realization preserves finite products. – David Carchedi Mar 26 2010 at 0:40

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Ok, here it is: "On the product of paracompact spaces" by Kiiti Morita. A link is here: journalarchive.jst.go.jp/english/… It is stated in a bit more generality, however, the case of a CW complex is mentioned on the first page. – David Carchedi Mar 26 2010 at 16:14

Space whose product with paracompact space is paracompact

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