Let $X$ be a set, $\tau_1 \leq \tau_2$ two comparable topologies on $X$ ($\tau_1$ is weaker than $\tau_2$) and consider some topological property $\varphi$ that holds for both $\tau_1$ and $\tau_2$. I am interested in a list of properties $\varphi$ that hold for all topologies $\tau$ in between $\tau_1$ and $\tau_2$, i.e. $\tau_1 \leq \tau \leq \tau_2$ and also in a list of properties for which there exists a topology $\tau$ in between $\tau_1$ and $\tau_2$ that does not satisfy $\varphi$.
- if $\varphi$ is Hausdorffness and $\tau_1$ satisfies $\varphi$ (and $\tau_2$ the discrete topology) then all $\tau \geq \tau_1$ satisfy $\varphi$. This is also true for many other separation axioms.
- if $\varphi$ is (sequential) compactness and $\tau_2$ satisfies $\varphi$ (and $\tau_1$ the trivial topology) then all $\tau \leq \tau_2$ satisfy $\varphi$.
- if $\varphi$ is first-countability or sequentiality and $X$ is infinite then $\varphi$ is not preserved in between the trivial topology and the discrete topology (e.g. the Arens-Fort space).
Does anyone know of a reference for such a list of "non-boring" properties $\varphi$ and suitable (e.g. maximal) choices for $\tau_1$ and $\tau_2$? (I think also of more advanced questions like: "what property $\psi$ does $\tau_1$ and $\tau_2$ has to satisfy such that if $\tau_1$ and $\tau_2$ satisfy $\varphi$ (e.g. are metrizable) then any topology in between also satisfies $\varphi$.)