For which $X$ is $X\times I$ collectionwise normal?

Question

Many normality-type properties can be characterised in terms of products with the unit interval $I=[0,1]$. For instance, if $X$ is a Hausdorff space, then;

1. $X$ is normal and countably paracompact if and only if $X\times I$ is normal.
2. $X$ is paracompact if and only if $X\times I$ is paracompact.
3. $X$ is perfectly normal if and only if $X\times I$ is hereditarily normal if and only if $X\times I$ is perfectly normal.
4. $X$ is stratifiable if and only if $X\times I$ is monotonically normal.

Of these statements, the first is due to Dowker, the second is obvious, the third is due to Katětov, and the last is due to Heath, Lutzer, and Zenor.

What is the class of those Hausdorff spaces $X$ for which $X\times I$ is collectionwise normal? For which members of this class is $X\times I$ hereditarily collectionwise normal?

Call the larger class defined here $\mathcal{C}$ and let $\mathcal{C}'$ be the more exclusive class. Clearly each metrisable space belongs to $\mathcal{C}'$. From the above we have more generally that each stratifiable space belongs to $\mathcal{C}'$. Each paracompact space belongs to $\mathcal{C}$, but need not belong to $\mathcal{C}'$. Clearly each member of $\mathcal{C}$ is collectionwise normal. However, the inclusion is strict, since Balogh has constructed a hereditarily collectionwise normal space $X$ for which $X\times I$ is not even normal.

Is there a nonparacompact space in $\mathcal{C}$?

Of the class $\mathcal{C}'$, each of its members is perfectly normal and hereditarily collectionwise normal. On the other hand, $\mathcal{C}'$ is not the class of hereditarily paracompact spaces, since there are examples of such which are not perfectly normal and consequently whose product with $I$ is not hereditarily normal.

Is every member of $\mathcal{C}'$ hereditarily paracompact?

Answer 1

The first part of the first question was in fact answered in the mid 1950s.

Theorem (Dowker): If $X$ is a countably paracompact and collectionwise normal $T_2$ space and $Y$ is compact and metrisable, then $X\times Y$ is collectionwise normal. $\quad\blacksquare$

This comes from C. Dowker, Homotopy Extension Theorems, Proc. London Math. Soc., 6, (1956), p.100–116.

Thus, while a necessary condition for $X\times I$ to be collectionwise normal is that $X$ be collectionwise normal and countably paracompact, it is also sufficient. The first uncountable ordinal $\omega_1$ in the order topology is an example of a collectionwise normal, countably paracompact space which is not paracompact. Thus $\mathcal{C}$ does indeed contain a nonparacompact space.

As for the second question, as to when $X\times I$ is hereditarily collection collection normal, I have not yet made progress.

• The Michael line $\mathbb{R}_M$ is monotonically normal, so hereditarily collectionwise normal and hereditarily countably paracompact. It is hereditarily paracompact, but not perfectly normal. The product $\mathbb{R}_M\times I$ is paracompact, but not hereditarily normal.
• The Sorgenfrey line $\mathbb{R}_\ell$ is monotonically normal and perfectly normal. The product $\mathbb{R}_\ell\times I$ is hereditarily paracompact. Since $\mathbb{R}_\ell$ is not stratifiable, $\mathbb{R}_\ell\times I$ is not monotonically normal.
• The ordinal space $\omega_1$ is monotonically normal, but neither perfectly normal nor paracompact. The product $\omega_1\times I$ is collectionwise normal, but neither paracompact nor hereditarily normal.