Show that the intersection of two topologies on the same set X is also a topology on X, but that their union may or may not be a topology.

I think this is way easier than i think. I have tied my mind in knots.

You need to check (T1-T3)

(T1) Obv from def of topology

(T2)$\displaystyle T_i \cap T_j = ( T_{11} \cap T_{12}) \cap (T_{21} \cap T_{22}) $ where $\displaystyle T_{11}, T_{12} \in T_i, T_{21}, T_{22} \in T_j$

$\displaystyle = ( T_{11} \cap T_{21}) \cap (T_{12} \cap T_{22}) $

$\displaystyle = \in T_i, \in T_j$

I cannot see how to get (T3) to work