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;

- $X$ is normal and countably paracompact if and only if $X\times I$ is normal.
- $X$ is paracompact if and only if $X\times I$ is paracompact.
- $X$ is perfectly normal if and only if $X\times I$ is hereditarily normal if and only if $X\times I$ is perfectly normal.
- $X$ 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?