Let $Y$ be a topological space and $V\subset Y\times Y$. For each $y\in Y$, we define $$V[y]=\{z\in Y\,:\,(y,z)\in V\}.$$ It is possible to show that, if $U\subset Y\times Y$ is an open set and $\Delta\subset U$ (where $\Delta=\{(y,y)\in Y\times Y\,:\, y\in Y\}$), then $$\Delta(U):=\{U[y]\,:\,y\in Y\}$$ is an open covering of $Y$. Using this, I want to prove the following
Let $Y$ be a paracompact topological space and $U\subset Y\times Y$ open with $\Delta\subset U$. Show that exists an open set $V\subset Y\times Y$, with $\Delta\subset V$, such that $$\bigcup_{y\in Y}(V[y]\times V[y])\subset U.$$
Using the fact above, I know that $\Delta(U)$ is an open covering of $Y$. Since $Y$ is paracompact, there exists, by definition, an open neighborhood-finite refinement for $\Delta(U)$, say $\scr U$, which means that, exists $\scr U$ such that
- every element of $\scr U$ is an open set in $Y$;
- $\scr U$ is a covering of $Y$ (i.e. $\displaystyle \bigcup_{W\in \scr U}W=Y$) and, for each element $W\in \scr U$, there exists some element $U[y]\in \Delta(U)$ such that $W\subset U[y]$ (refinement condition); and
- $\scr U$ is neighborhood-finite: for each $y\in Y$, there is a neighborhood $N$ of $y$ such that $\{W\in \scr U$$\,:\, W\cap N\neq \varnothing\}$ is finite.
But, what path to follow? How, from this, do I construct the required set $V$?