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?



How to choose a continuous function which vanishes **only** on the closed set
1. The function $f:\bar X\times \bar X\to\mathbb R$ exists if and only if the compactification $\bar X$ of $X$ is metrizable, which implies that the locally compact space $X$ is metrizable and separable.


This follows from a well-known metrizability theorem saying that a compact topological space $K$ is metrizable if and only if the diagonal is a closed $G\_\delta$-set in $K\times K$.


2. The function $g:\bar X\times \bar X\to \mathbb R$ exists if and only if the set $X\times X$ is of type $F\_\sigma$ in $\bar X\times \bar X$, which happens if and only if the locally compact space $X$ is $\sigma$-compact if and only if $X$ is Lindelof.


