We are reading John Roe's book *Lectures on Coarse Geometry*. We come across a question in P27 line 9:
Suppose $X$ is a paracompact and locally compact Hausdorff space, $\bar{X}$ is a compactification of $X$, how to use Urysohn's Lemma to choose continuous functions $f,g:\bar{X}\times\bar{X}\to \mathbb{R}^+$ such that $f$ vanishes **only** on the diagonal of $\bar{X}\times\bar{X}$, and $g$ vanishes **only** at infinity (that is, on $\bar{X}\times\bar{X}\backslash X\times X$)?

If $\bar{X}\times\bar{X}$ is perfectly normal, or the diagonal and the infinity are both $G_\delta$ sets, we can have that according to P213 in Munkres's book *Topology*. But we fail to verify the condition(we are not familiar with paracompact Hausdorff space).

Or any other way?