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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 $X\times X$.

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Welcome to MSE Heidi. Please read the FAQ. You need to say what you've done towards a problem in order to get help, not just state it. – Alexander Gruber Nov 4 '12 at 6:24
Are you sure that the last sentence is correct? I suspect that you want to prove that the topology on $Y\times Y$ induced by $q$ is the same as the product topology on $Y\times Y$, which isn’t what you actually wrote. – Brian M. Scott Nov 4 '12 at 6:27
I think you mean: Prove that the topology on $Y \times Y$ is the same as .... This is answered below, and is also relevant to – Ronnie Brown Nov 4 '12 at 17:41
oops! yes, i meant the topology on $Y \times Y$ as a product of the quotient topologies on $Y$ is the same... I feel like it intuitively makes sense but I'm not sure how to start a formal proof of it – Heidi Nov 5 '12 at 9:32

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.

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