Help understanding proof of every topological group is regular
Solution 1:
If $y\in V\cap (X - U)W$, then $y\in V$ and $y = st$ for some $s\notin U$ and $t\in W$. Then $(y,t)\in V\times W$, so $f(y,t) \in U$, i.e., $yt^{-1}\in U$. On the other hand, $s = yt^{-1}\in U$. This contradiction shows $V\cap (X - U)W = \emptyset$.