$(X \times X) /{\sim'}\cong (X/{\sim}) \times (X/{\sim})$

Question

The full description of this problem is:

Let $X$ be a topological space. Let $\sim'$ be the equivalence relation on $X\times X$ defined by $(x,y)\sim'(x',y')$ iff $x \sim x'$ and $y \sim y'$

Prove that $(X \times X) /{\sim'}\,\cong (X/{\sim}) \times (X/{\sim})$.

Intuitively it makes so much sense, but I am really stuck on how to write a formal proof. Can anyone give me a outline?

Great thanks!

Answer 1

Hint

The homeomorphism is given by $$f:(X\times X)/{{\sim}'}\longrightarrow (X/{\sim})\times (X/{\sim})$$ defined by $$f([(x,y)])=([x],[y]).$$

Prove it !

Answer 2

Hint -

Using the universal property of quotient spaces you can get a continuous map $\phi:(X\times X)/ {\sim'}\to X/ {\sim}\times X/ {\sim}$ which in this case you can show to be bijective and open.