pi-base
Property: Strongly zero-dimensional spaces
A property that isn't on pi-base is that of strongly zero-dimensional spaces.
Those are spaces $X$ for which the Lebesgue covering dimension or equivalently large inductive dimension is $0$.
Equivalently, and I think this should be the definition on pi-base, a space $X$ is strongly zero-dimensional or ultranormal if for any closed disjoint $E, F\subseteq X$ there is clopen $U\subseteq X$ with $E\subseteq U, F\subseteq U^c$.
Contrast with zero-dimensional spaces, for which small inductive dimension of $X$ is $0$.
If we add this property the theorem [regular + P-space => zero-dimensional] on pi-base could be strengthened by replacing it with [regular + P-space => strongly zero-dimensional] as in this paper https://www.sciencedirect.com/science/article/pii/0016660X72900268
We could also add [strongly zero-dimensional + $T_1$ => zero-dimensional] and [strongly zero-dimensional + regular => zero-dimensional] See Charalambous's Dimension Theory
If we also add the property of ultrametrizable spaces, we could add those theorems: [metrizable + strongly zero-dimensional => ultrametrizable] [ultrametrizable => strongly zero-dimensional] [ultrametrizable => metrizable] A space is called ultrametrizable if it can be given an ultrametric i.e. metric $d$ such that $d(x, y)\leq \max(d(x, z), d(z, y))$
The following converse to strong zero-dimensionality could be added: [zero-dimensional + Lindelof => strongly zero-dimensional] Proof: If $X$ is Lindelof then $\text{dim }X \leq \text{ind } X$ where $\text{dim}$ is the Lebesgue covering dimension. If $\text{ind }X = 0$ then $\text{dim }X = 0$ so $X$ is strongly zero-dimensional (as I mentioned before, $\text{dim }X = 0$ iff $\text{Ind }X = 0$ where $\text{Ind}$ is the large inductive dimension).
Another one is [normal + extremally disconnected => strongly zero-dimensional]
