# CCC +　collectionwise normality =>　paracompact?

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 $$s \in S$$.

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 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 ?

• 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