Example on a product topology

Question

There is a proposition that states: The product topology is the weakest topology on $X$ that makes each projection $P_i (i \in I)$ continuous.

So I was searching for an example of a product topology where the projections of $X$ aren't continuous, but I didn't find any yet. Can anyone in finding an example please?

Answer 1

So I was searching for an example of a product topology

I assume you've meant "an example of a topology on product". There is no "other" product topology, there is only one.

Anyway take the trivial topology on $X\times Y$ (meaning only $\emptyset$ and $X\times Y$ are open) and assume that $X$ has a proper, nonempty open subset $U$ (i.e. the topology on $X$ is not trivial). Let $\pi:X\times Y\to X$ be the projection. Is $\pi^{-1}(U)$ open?