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 $\varphi$.)



Preservation of topological properties in between two topologies
[**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](https://wrbrian.files.wordpress.com/2012/01/refinements1.pdf) [*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](https://mathoverflow.net/questions/87834/continuous-bijections-vs-homeomorphisms/203126#203126).]
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).


