Show that $X \times Y$ is homeomorphic to $Y \times X$

Question

This seems to be a trivial result, I just wish to check for correctness.

Show $X \times Y$ is homeomorphic to $Y \times X$

Let $\tau$ be the topology on $X$, $j$ be the topology on $Y$

Then a homeomorphism is given by $f: X \times Y \to Y \times X, (a,b) \to (b,a)$

We can show that $f$ is a homeomorphism by showing that it is continuous and open.

$f$ is continuous because given $B \times A$, $f^{-1}(B \times A) = A \times B$ is open in the product topology on $X \times Y$.

Similarly, we can show that $f$ is open in that $f(A \times B) = B \times A$ is open in the product topology on $Y \times X$

Therefore $f$ is a homeomorphism.

Is this correct?

Answer 1

I think that it the idea is correct. Maybe one should add that on $X\times Y$ and $Y\times X$ we choose the product topology. The basis of such product topology is the collection $A\times B$ with $A$ and $B$ open in the respective spaces (or in the respective bases). In order to have continuity it is enough to check the "open set property" for the basis elements.