Hello,

I'm having trouble with this problem:

Suppose $\displaystyle X$ is a paracompact topological space. Let $\displaystyle U$ be an open set of $\displaystyle X\times [0,\infty)$ which contains $\displaystyle X\times\{0\}$. Show that there is a map $\displaystyle f\colon X\rightarrow (0,\infty)$ such that $\displaystyle y\leq f(x)$ implies $\displaystyle (x,y)\in U$.

Note that all my paracompact spaces are assumed to be Hausdorff.

I know that paracompact spaces are in particular normal, and I tried to use an Urysohn function, but I couldn't find one with the right property.

I also attempted a partition of unity, but the codomain of f is not supposed to contain 0, which throws me off.