TITLE: Different versions of Urysohn's Lemma
QUESTION [5 upvotes]: In short, I encountered two different versions of Urysohn Lemma, which I shall present here:
Version 1: [Big Rudin, 2.12, page 39]: Suppose X is a locally compact Hausdorff space, V is open in X, $K \subset V$ and K is compact. Then there exists an $f \in C_c(X)$ such that $K \prec f \prec V$ with the following notation:

$f \in C_c(X)$ if f is continuous and has compact support
$K \prec f$ if $0 \le f(x) \le 1$ for all x and $f(K) = 1$
$f \prec V$ if $0 \le f(x) \le 1$ for all x and the support of f lies in V.

Version 2: [Munkres, Topology, page 201]: Let X be a normal space, let A and B be disjoint closed subsets of X. Let [a,b] a closed interval in the real line. Then there exists a continuous map $f: X \rightarrow [a,b]$ such that $f(x) = a \forall x \in A$ and $f(x) = b \forall x \in B$
My question is that which version is "stronger"? I've tried to deduce whether one version is the consequence of another, but it seems not the case. As far as I read through both books, each version has distinct applications, so how the difference in the initial condition affect their application?

REPLY [4 votes]: The theorem in Munkres is the one usually referred to as Uryso(h)n’s lemma; I consider Rudin’s use of the term incorrect. The two results are not directly comparable: a locally compact Hausdorff space need not be normal, and a normal space need not be locally compact.

Added: However, as Andreas Blass reminds me in the comments, one can deduce Rudin’s result from Uryson’s lemma by passing to the one-point compactification $X^*$ of a locally compact Hausdorff space $X$: it’s a compact Hausdorff space and therefore normal, and a compact subset of $X$ is still closed in $X^*$.

Answers to this question give some examples of locally compact Hausdorff spaces that are not normal; $\Bbb Q$ is a familiar example of a normal (indeed metrizable) space that is not locally compact. The space of irrationals is another.