# Is there a CCC and collectionwise normal space, that isn't paracompact? [closed]

Is there a CCC and collectionwise normal space, that isn't paracompact?

As we know, CCC + monotone normality $\implies$ Lindelöf.

CCC + collectionwise normality $\implies$ paracompact?

• Collectionwise normality: if $X$ is a $T_{1}$ space and for every discrete family $\{F_{s}\}_{s \in S}$ of closed subsets of $X$ there exists a discrete family $\{V_{s}\}_{s \in S}$ of open subsets of $X$ such that $F_{s}$ $\subset$ $V_{s}$ for every s $\in$ S.

Edit: (t.b.) also posted on MO

-
I happen to know all the terms here except "monotone". Most of these terms are rather specialized items in point-set topology. Would it be too much trouble to add the definitions (at least of "collectionwise normal" and "CCC") for the benefit of the readers? –  t.b. Oct 24 '11 at 11:18
OK, no problem. –  Paul Oct 24 '11 at 11:34
I posted an answer on the MO forum. No need to repost, I suppose? –  Henno Brandsma Oct 24 '11 at 20:49
Just an extra question: counterexamples exists, as per my post on MO, but what about first countable ones? My examples are Fréchet-Urysohn spaces, but not first countable (in this form that cannot be). –  Henno Brandsma Oct 24 '11 at 20:56
@Henno: I think it is better to keep the answer in one place, thus I've voted to close this question here. Those interested have the link anyway. –  t.b. Oct 24 '11 at 23:26
add comment

## closed as too localized by t.b., Asaf Karagila, Ｊ. Ｍ., Mike Spivey, DidOct 31 '11 at 13:31

This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.