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 $s \in S$.


1 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 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 ?

  • $\begingroup$ Is the Cp(L(ℵ1)) not paracompact? I cannot find the reference, so if it is possible give a bit more details on that. $\endgroup$
    – Rnst
    Oct 28, 2013 at 21:58
  • $\begingroup$ added an argument... $\endgroup$ Oct 30, 2013 at 8:30

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