Q:

Equivalence (Or non-equivalence) of definitions for Metric Spaces

So, I've stumbled across a definition of a metric space that has given me some pause. The definition of a Metric Space that I've always used has been the following: 

A Metric Space is a set $X$ together with a function $d: X \times X \to \mathbb{R}$ satisfying the standard metric axioms. 

However, while reading Munkres' Topology, he presents the following definition of a metric space:
A Metric Space is a metrizable space $X$ together with some specific metric $d$ that gives the topology of $X$. 
The reason this definition has given me pause is not only the fact that I've never seen it but that the first definition of a metric space has no assumption that $X$ must be metrizable. In fact, it doesn't even assume $X$ is a topological space. Now, of course, with a metric $d$ on $X$, we may equip $X$ with the metric topology induced by $d$ and make $X$ a topological space, but it seems to me each of the two definitions require us to know quite different things about $X$ before calling it a metric space. The first definition seems more "bare-bones" in a sense while the second requires us to know that $X$ possesses certain kinds of structures/properties. Are these two definitions, in fact, equivalent, or if they are inequivalent, are there other reasons at play for desiring an alternative notion of a metric space?

A:

This community wiki solution is intended to clear the question from the unanswered queue.
Your question has been answered in Lee Mosher's second comment.
The difference is that a metric space in the standard definition is a pair $(X,d)$ with a set $X$, whereas in Munkres' definition it is a pair $(X,d)$ with a topological space $X$. As Lee Mosher remarked, in the second case one should more precisely write it as a triple $(X, \mathfrak T,d)$ with a set $X$ and a topology $\mathfrak T$ on $X$.
There is a $1$-$1$-correspondence between "standard pairs" and "Munkres triples". In fact, the functions
$$(X, \mathfrak T,d) \mapsto (X,d), \\(X,d) \mapsto (X, \mathfrak T_d,d) ,$$
where $\mathfrak T_d$ is the metric topology generated by $d$, are inverse to each other. It is therefore a matter of taste which definition you prefer.
Perhaps Munkres intention is to focus on the concept of a metrizable space. This is a space, not a set, with a certain property. This might be also the reason why he explicitly says that a metric space is a metrizable space together with some specific metric $d$ that gives the topology of $X$. The word "metrizable" could of course be omitted.

