Suppose we have two topological spaces X, Y, and we construct from them the product topology. Now suppose that A is closed in A and B is closed in Y. I want to prove that $A\times B$ is closed in $X \times Y$.
Here's what I've done so far
Consider the compliment
$(X\times Y)/ (A\times B)=((X/A)\times Y)\cup (X \times (Y/B))$.
As A is closed in X$\Rightarrow X/A$ is open same goes for $Y/B$.
Now I now that once I can say the union is of open sets and so open we'll be done my question however is :
How can we decide that say $(X/A \times Y)$is open , obviously not all sets in Y are open as B is in Y, so how can we decide that Y is open ?
By the definition of product topology, a the sets $U_A \times U_A$ for $U_A\subseteq A$ (open) and $U_B \subseteq B$(open) form a basis of product $A \times B$. In particular, $X- A$ is open in $X$ and $Y$ is open in $Y$.