This is a question from a practice qualifying exam.
Let $X, Y, T$ be topological spaces. Define $p_X: X\times Y\to X$ and $p_Y:X\times Y\to Y$ to be the projection maps. Then $f:T\to X\times Y$ is continuous if and only if the compositions $p_X \circ f : T\to X$ and $p_Y\circ f: T\to Y$ are continuous. (I have already proven this).
Prove that the product topology on $X\times Y$ is the unique topology that, for all spaces $T$ and functions $f$, has the previous property.
As shown here: Product topology on $X \times Y$ the smallest topology when $f(x, y) = x$ and $g(x, y) = y$ are continuous functions?, we know that any topology n $X\times Y$ with the property must at least contain the product topology (just use $T = X\times Y$ and $f = id$). Thus, the only thing that I need to show is that any topology for which the property holds is contained in the product topology, but unfortunately I'm not really sure how to approach this.
First, note that if you have any topology $\tau$ on $X\times Y$ with this property, then the projections $p_X$ and $p_Y$ are both continuous with respect to $\tau$ (take $T=(X\times Y,\tau)$ and $f=id$). Now if $\tau_1$ and $\tau_2$ are two topologies on $X\times Y$ with this property, then the identity map $X\times Y\to X\times Y$ is continuous from $\tau_1$ to $\tau_2$ (use the property for $\tau_2$ and take $T=(X\times Y,\tau_1)$, $f=id$). By symmetry, the identity map is also continuous from $\tau_1$ to $\tau_2$. Thus $\tau_1=\tau_2$.