Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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

As we know, CCC + monotone normality => lindelof.

CCC + collectionwise normality => paracompact?

CCC = countable chain condition

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.

share|improve this question
    
3  
math.stackexchange.com/questions/75359 Please do not post on multiple sites, and if you do it please link to the other questions, at least. –  Theo Buehler Oct 24 '11 at 17:33

1 Answer 1

up vote 8 down vote accepted

Yes, there is. Let $I = \omega_1$ be the first uncountable ordinal, and let $P = \{0,1\}^I$ be the uncountable product of discrete spaces of 2 points. Let $S$, the so-called $\Sigma$-product be its subspace of all points that have at most countably many coordinates different from $0$.

It is well known that $S$ is ccc (as a dense subset of a ccc space $P$) and countably compact (but not compact, being dense in $P$) and (hereditarily) collectionwise normal, but not paracompact (being countably compact and non-compact). Proofs of some of these facts can be found here, e.g.

Corson showed in this paper (cannot find free download) that if $X$ is dense in a product of metrizable spaces, and $X \times X$ is normal, then $X$ is collection wise normal. This can be used to show the collectionwise normality, as $S \times S$ is homeomorphic to $S$, so one only needs to show normality.

A very related example is the set $C_p(L(\aleph_1))$, where $L(\aleph_1)$ is the one-point Lindelöfication of a discrete space of size $\aleph_1$ (add a point $\infty$ with co-countable neighbourhoods), and $C_p(X)$ is the space of continuous real-valued functions on a space $X$, in the subspace topology of $\mathbb{R}^X$. This example is discussed on page 113 of the book General Topology III, in the Encyclopedia of Mathematical Sciences series (volume 51). All spaces of the form $C_p(X)$, for Tychonoff $X$, are ccc, and if they are normal, they are collectionwise normal (due to Reznichenko), so it's natural to look for examples there.

[added:] This space is not paracompact because for a ccc space like $C_p(X)$ paracompact is equivalent to being Lindelöf, and $C_p(L(\aleph_1))$ contains the $\Sigma$-product of copies of $[0,1]$ as a natural closed subspace (take all $f$ with all values in $[0,1]$ and $f(\infty)=0$), and this $\Sigma$-product, like the one mentioned above, is ccc and countably compact, but not compact (so not Lindelöf, and thus not paracompact).

As an aside: by well-known results, both these spaces are Fréchet-Urysohn, but not first countable. Can there be first countable examples ?

share|improve this answer
    
Is the Cp(L(ℵ1)) not paracompact? I cannot find the reference, so if it is possible give a bit more details on that. –  Rnst Oct 28 '13 at 21:58
    
added an argument... –  Henno Brandsma Oct 30 '13 at 8:30

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.