Is $(X,\mathcal T)$ a $T_0$-space? 
Let $(X,\mathcal T_1)$ and $(X,\mathcal T_2)$ be topological spaces. Now define $\mathcal T=\mathcal T_1 \cap \mathcal T_2$.

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*If $(X,\mathcal T_1)$ and $(X,\mathcal T_2)$ are $T_1$-spaces, is $(X,\mathcal T)$ a $T_1$-space?


*If $(X,\mathcal T_1)$ and $(X,\mathcal T_2)$ are $T_0$-spaces, is $(X, \mathcal T)$ also $T_0$?

For the first question, I have tried the following:
Let $x \in X$, then $X\setminus \{x\} \in \mathcal T_1$ and $X\setminus \{x\} \in \mathcal T_2$, since $\{x \}$ is closed in $(X,\mathcal T_1)$ and $(X, \mathcal T_2)$. That is, $X\setminus \{x \} \in \mathcal T$, and so $\{x \}$ is closed in $(X,\mathcal T)$. Since $x$ was chosen arbitrarily, we have that $(X, \mathcal T)$ is a $T_1$-space.
As for the second bullet, I am not quite sure. I am thinking that the statement is not true, but I cannot think of a suitable counter-example. Can anyone please help me?

Hint: To find a counterexample for the second question, try considering topologies on a set $X$ which has two points.
