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Let $(X,\tau)$ be a topological space and let $\sim$ be an equivalence relation on $X$. Now define an equivalence relation $\approx$ on $X \times X $ by $ [(x,y)]_\approx = [x]_\sim \times[y]_\sim $

Is it true that $X/\sim \times $ $X/\sim $ $ \cong (X\times X)/\approx $ ?

MCL
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1 Answers1

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$[x]_\sim \times [y]_\sim$ and $[(x,y)]_\approx$ as sets are bijective. In one case you first take the projection to the cosets, and then the product. In the other you first take the product and then project to the cosets defined with the above equivalence independantly on the two components.

I'm assuming you want to check if the induced topology is the same.

It is straightforward that any open set in $\tau$ is mapped to the "same" set in the two spaces which is how their topology is induced therefore they are homeomorphic.

Mr.P
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  • Regarding the first paragraph: the sets $[x]{∼} × [y]{∼}$ and $[(x, y)]{≈}$ are equal (by definition). I suspect you mean instead that $(X/{∼}) × (Y/{∼})$ and $(X × Y)/{≈}$ are bijective. Regard the third paragraph: the resulting spaces are _not always homeomorphic. Your ‘argument’ doesn’t make sense, since the quotient topology and product topology are not defined by taking images of open sets. – Jendrik Stelzner Jan 15 '23 at 11:55
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    @JendrikStelzner can you provide a proof that they're not always homeomorphic or a source? From my digging, all people do is show that the natural map between them is not a homeomorphism, not that they are not homeomorphic – Jakobian May 17 '24 at 16:01