# The product of quotient spaces is a quotient space

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 $X/\sim \times$ $X/\sim$ $\cong (X\times X)/\approx$ ?

$[x]_\sim \times [y]_\sim$ 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.
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.