This question might be silly, however I'd like to be $100\%$ sure.
Suppose we have some topology $\tau$ on a set $X$ which gives us the $T_i$ space for some $i<6$. It's totally possible to improve topological properties (here I'd like to improve separation axioms) - simply consider basic anti-discrete topology, which consists only of $\emptyset$ and $X$.
My question is about the opposite situation. Is it possible to have topology $\tau_1$ on set $X$ and a bigger topology $\tau_2 \supset \tau_1$, such that $(X,\tau_1)$ is $T_i$ space while $(X,\tau_2)$ does not fulfill $T_i$ space axiom.
I'm pretty sure such situation is not possible, but if it is - please provide me with a proper counterexample.
For $i\leqslant 2$ I'm sure it's not the case, I wonder if making topology larger (which at the same time introduces new closed sets) can prevent $(X,\tau)$ from being $T_i$ space.