Let $X$ be a topological space, $p:X\to Y$ be a quotient map, and $q:X\times X\to Y\times Y$ be the quotient map defined by $q(x,y)=(p(x),p(y))$. Prove that the topologies on $Y$ is the same as the topology on $Y\times Y$ as a quotient of the product topology on $X\times X$.
The suggested result is false: the example given in the book Topology and Groupoids,(T&G) p.$111$, is actually that if $p: \mathbb Q \to Y$ is the quotient map identifying all of $\mathbb Z$ to a single point, then $p \times 1: \mathbb Q \times \mathbb Q \to Y \times \mathbb Q$ is not a quotient map. So it is expected that $p \times p$ is not a quotient map.
The result is true for $p: X \to Y$ if $X$ and $Y$ are locally compact and Hausdorff.
This problem led me in my 1961 Oxford thesis to propose using the category of Hausdorff $k$-spaces, and this proposal has been improved by using what have been called compactly generated spaces, i.e. spaces $X$ which have the final topology with respect to (a set of) continuous maps $C \to X$ for $C$ compact Hausdorff. This is explained in Section 5.9 of T&G. See also this ncatlab link on convenient categories of topological spaces.