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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.2012-11-04
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    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.2012-11-04
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    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 http://math.stackexcnge.com/questions/316972012-11-04
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    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 it2012-11-05
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    My previous comment should have referenced http://math.stackexchange.com/questions/316972018-12-02

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