In the co-finite topology on $\mathbb{R}$, every singleton $\{x\}$ is closed (its complement is co-finite = open). This makes it a $T_1$-space (for any two distinct points, each has a neighbourhood not containing the other). However, any two non-empty open sets have non-empty intersection (both are co-finite sets, whose intersection is also co-finite, hence non-empty). So the Hausdorff ($T_2$) separation axiom fails. It is $T_1$ but not $T_2$.