pi-base
Property Suggestion: Strongly collectionwise normal
Property Suggestion
A space is said to be strongly collectionwise normal or divisible (name divisible already exists for other properties related to cleavability of spaces) provided that for each neighbourhood $U\subseteq X\times X$ of the diagonal $\Delta_X = {(x, x) : x\in X}$ of $X\times X$ there exists a neighbourhood $V$ of $\Delta_X$ such that $V\circ V\subseteq U$.
Rationale
This property appears in the textbooks General topology by Kelley (in a conjecture, not named explicitly; Kelley references a result of Cohen) and papers Strong Collectionwise Normality and M. E. Rudin's Dowker Space by K. P. Hart, for example.
The property has a particular form for completely regular spaces: A completely regular space $X$ is strongly collectionwise normal if and only if the family of all neighbourhoods of the diagonal is the largest admissible uniformity on $X$.
Relationship to other properties
Every strongly collectionwise normal space is collectionwise normal. Every fully normal space is strongly collectionwise normal.
Note that example 207 from the above PR could be improved in the future if we were to add the property monotonically normal to pi-base, since its an example of a monotonically normal space which isn't strongly collectionwise normal. On the other extreme end, there exist countable $T_3$ space which isn't monotonically normal, see here. In particular, even if we demand that every subspace of $T_1$ space $X$ is fully normal, this still wouldn't imply that $X$ is monotonically normal.
This shows that the property fully normal and monotonically normal are pretty much incomparable with each other.
Here is a diagram I made. It shows that all the properties we are interested it are in fact not reversible.
The addition of uncountable $\Sigma$-product of $\omega$ would be good as an example.
Very useful diagram. I am planning to add hereditarily collectionwise normal and monotonically normal very soon.
