Property Suggestion
A space is said to be Collectionwise Hausdorff (CWH) provided, for any closed discrete $Y$, there is a family of pairwise disjoint open sets, with each point in $Y$ being contained in exactly one of the open sets.
One can also study the hereditary version of this property (assuming $T_1$, this is equivalent to the above criterion where $Y$ is required to only be discrete, not closed), but I don't know of any theorems that can be achieved using it instead of the normal version, so there is little need.
Rationale
This property seems to have been studied more by set theorists than "pure topologists". I encountered it in "Remarks on $\aleph_1$-CWH not CWH first countable spaces." (zbMATH 0847.54005); it is also mentioned in "PFA(S)[S] for the masses." (zbMATH 1380.54005) and "The Density Topology" (zbMATH 0305.54039).
I figured it would be appropriate to add to pi-base since pi-base already features collectionwise normal (CWN), as well as variations such as hereditarily CWN and strongly CWN, and CWH forms a natural weakening of it.
Relationship to other properties
Every $T_1$, CWH space is $T_2$.
Every CWN space is CWH.
Every weakly countably compact, $T_2$ space is CWH.
Every CWH, ccc space has countable extent.
The weaker "every CWN, ccc space has countable extent" cannot be deduced yet.
It is consistent with, but not provable in, ZFC that every first countable, normal space is CWH.
Property Suggestion
A space is said to be Collectionwise Hausdorff (CWH) provided, for any closed discrete$Y$ , there is a family of pairwise disjoint open sets, with each point in $Y$ being contained in exactly one of the open sets.
One can also study the hereditary version of this property (assuming$T_1$ , this is equivalent to the above criterion where $Y$ is required to only be discrete, not closed), but I don't know of any theorems that can be achieved using it instead of the normal version, so there is little need.
Rationale
This property seems to have been studied more by set theorists than "pure topologists". I encountered it in "Remarks on$\aleph_1$ -CWH not CWH first countable spaces." (zbMATH 0847.54005); it is also mentioned in "PFA(S)[S] for the masses." (zbMATH 1380.54005) and "The Density Topology" (zbMATH 0305.54039).
I figured it would be appropriate to add to pi-base since pi-base already features collectionwise normal (CWN), as well as variations such as hereditarily CWN and strongly CWN, and CWH forms a natural weakening of it.
Relationship to other properties
Every$T_1$ , CWH space is $T_2$ .$T_2$ space is CWH.
Every CWN space is CWH.
Every weakly countably compact,
Every CWH, ccc space has countable extent.
The weaker "every CWN, ccc space has countable extent" cannot be deduced yet.
It is consistent with, but not provable in, ZFC that every first countable, normal space is CWH.