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.