Suppose that $(X,\tau)$ is a $T_0$-topological space without isolated point. As it is well-known, all cofinite subsets together with the empty-set forms a topology on $X$ which is a $T_1$ topology on $X$ and we call it the cofinite topology and denote it by $(X,\tau_f)$. Now $(X,\tau\vee\tau_f)$ is another topological space. Can someone help me to prove our disprove the following statement?
If $U\in \tau\vee\tau_f$ but $U\notin \tau$, then $U$ has finite complement.
The topology $\tau\vee\tau_f$ is defined to be the smallest topology which contains both $\tau$ and $\tau_f$.