Preservation of topological properties in between two topologies

Question

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 $\varphi$.

Examples:

• 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 $\varphi$.)

Answer 1

[Edit: Originally my second point below stated incorrectly that if $\tau_1$ and $\tau_2$ are as described, then every topology in between is metrizable. This is not what the theorem in my paper says, but yesterday, in my haste, I thought that the theorem in my paper should imply this statement. It does not, as I now realize, and I've corrected the error.]

I explore some questions like this in a paper [Preservation and destruction in simple refinements, Top. App. 178 (2014), pp. 236-247]. Some of the main results along these lines are:

1. If $\tau_1$ is (completely) regular and $\tau_2 \supseteq \tau_1$, then every topology in between is (completely) regular if and only if every $\tau_2$-open set is the intersection of an open set and a closed set in $\tau_1$.

2. Suppose $\tau_1$ is metrizable and $\tau_2$ has a basis of the form $\tau_1 \cup B$, where $B$ is countable, and every member of $B$ is the intersection of an open set and a closed set in $\tau_1$. Then $\tau_2$ is also metrizable. However, unless $\tau_1 = \tau_2$, there will be non-metrizable topologies strictly in between the two. [The cited paper does not contain a proof of this latter fact, but you can read about it here.]

3. Suppose $\tau_1$ is Polish. Then the space with basis $\tau_1 \cup \{A\}$ is Polish if and only if $A$ is the intersection of an open set and a closed set in $\tau_1$.

4. Suppose $\tau_1$ is locally compact. Then the space with basis $\tau_1 \cup \{A\}$ is locally compact if and only if $A$ and its complement are each the intersection of an open set and a closed set in $\tau_1$.

Also, let me say that your comment about separation axioms is true for lower separation axioms ($T_{2\frac{1}{2}}$ and down) but not for higher ones ($T_3$ and up).