Topology on a product space, such that the projection is not continuous

Question

Can we find topological spaces $(X,T_1)$ and $(Y,T_2)$, and a topology on the product space $T$ : $(X\times Y,T)$ such that the projection $p :X\times Y \rightarrow X$, such that for all $x \in X$ and $y\in Y$ : $p(x,y)=x$, and $p$ is not continuous ?

Answer 1

The trivial topology $\{X\times Y,\emptyset\}$ has only one non-empty open set. Therefore, unless $T_1$ is trivial as well, that's an example (take the preimage of a non-trivial open set in $T_1$).

Answer 2

No! Its not possible, because the topology product is the minimum topology that the projection are continous.