Question

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.

Answer 1

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 compactification of a discrete space of size $\aleph_1$, 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 $C_p(X)$ are ccc, and if they are normal, they are collectionwise normal (due to Reznichenko), so it's natural to look for examples there.

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