Existence of a topology satisfying a certain condition This might be a really silly question but i'm curious to know if the following is true:
Let $X$ be an infinite set with topologies $\tau_1$ and $\tau_2$ such that $\tau_1\subset\tau_2$. Is it always possible to find a topology $\tau_3$ such that $\tau_1\subset\tau_3 \subset\tau_2$? If not then can some conditions be imposed on $X$ for which this will always hold true?
If at all we can impose some conditions on $X$, for which the above holds true, then we can obtain an infinite chain of topologies satisfying, $\tau_1\subset\tau_3 \subset\tau_4\subset...\subset\tau_2$
I worked out some examples, in some cases I could come up with such a topology and in some cases I couldn't. 

For $T_1$ spaces I found this paper.
It refers to a lot of older papers, some of which also study whether some topologies have immediate predecessors (topologies $\tau_1 \subsetneq \tau_2$ with no topology in-between). In seems to be a well-studied subject. 
So sometimes it's not possible, sometimes it is, but it's complicated stuff.
