I'm looking for an example to demonstrate that fact that the Nagata-Smirnov thm is "more useful" than Urysohn's. That is, I'm looking for a space that you can prove is metrizable using Nagata-Smirnov but not using Urysohn.
I've found (in Steen & Seebach's "Counterexamples") some examples of spaces which have $\sigma$-locally finite bases but are not second-countable. Problem is, they all have an explicit metric... In fact, most of them are only described as metric spaces, with no metric-independent description of the topology. So these examples are not very motivating...
I know there are theoretical implications of NG thm that provides motivation, but right now I'm looking specifically for a down-to-earth, tangible example, preferably one that the average undergrad can cope with.
Edit: here is the formulation of the thms:
Urysohn: A $T_{3}$, second-countable space (i.e. $X$ has a countable base for its topology) is metrizable.
Nagata-Smirnov: A topological space is metrizable iff it's $T_{3}$ and has a base for its topology that is a countable union of locally finite families. (where a family $A_i, i \in I$ of subsets of $X$ is locally finite in $X$ iff every $x \in X$ has a neighbourhood $O_x$ such that $\{i \in I: O_x \cap A_i \neq \emptyset\}$ is finite.)
Bing-Nagata-Smirnov's theorem (BNS) is a necessary and sufficient condition for being a metrisable space, while Urysohn's theorem is merely sufficient. Many other general criteria for metrisability have been proved on the basis of BNS, look at the Encyclopedia of General Topology (chapters on metrisation theorems). A classic application is the fact that a paracompact locally metrisable space is metrisable (which is often used in manifolds). So BNS is a convenient general tool.
It has been the basis of a quite extensive literature on metrisability, and the more specialised Urysohn's theorem rather less so. One of the few concrete examples of an application of Urysohn is that the unit ball in $X^\ast$ in the weak-star topology is metrisable when $X$ is a separable Banach space. Not a concrete example, but useful nonetheless. In that case the metric on that ball is easy enough to construct without Urysohn, but at least we know a metric exists, so even without knowing it we can apply sequential compactness of that ball (based on Alaoglu) instead of "mere" compactness.