Conditions for the product of two topologies to be a topology

Question

Take two topological spaces $(X,T_{X})$ and $(Y,T_{Y})$. What are the necessary conditions on $(X,T_{X})$ and $(Y,T_{Y})$ such that $$T_{1} := \{A \times B\mid A \in T_{X},B \in T_{Y}\}$$ ia a topology on $X\times Y$ and $$T_{2} := \{U \subseteq X \mid X \backslash U \text{ is infinite}\} \cup \{\emptyset,X\}$$ is a topology on $X$.

Answer 1

As discussed, if X is finite, $T_2$ is indiscrete.
If X is infinite, then $T_2$ is discrete because for
x in X, {x} is open since X - {x} is infinite.

Answer 2

The only way for $T_{1}$ to be a topology is if $T_{X}$ and/or $T_{Y}$ is the trivial topology. Anything else doesn't work.

The only way for $T_{2}$ to be a topology is if $X$ is finite.