In a product topology $X\times Y$, the open sets are defined as $V\times W$, where $V\in X$ and $W\in Y$ are open sets.
If $X\times Y$ is a topology, then the union of open sets should be open- i.e. $(A\times B)\cup (C\times D)=M\times N$ for some open sets $M\in X$ and $N\in Y$. We know that $(A\times B)\cup (C\times D)\neq (A\cup C)\times (B\cup D)$. How do we prove that the open sets $M$ and $N$ exist?
Thanks in advance!