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


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


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 $s \in S$.



CCC +　collectionwise normality =>　paracompact?
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](http://dantopology.wordpress.com/2009/12/12/a-note-about-sigma-product-of-compact-spaces/), e.g.


Corson showed in [this paper](http://www.jstor.org/pss/2372929) (cannot find free download) that if $X$ is dense in a product of metrizable spaces, and $X \times X$ is normal, then $X$ is collectionwise 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 ?


