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While reading a book on general topology I found the definition of a metric space as follows.

"A metric space (X,d) is a set X equipped with a metric d on X"

What exactly does it mean to say that a set is equipped with a function?

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up vote 7 down vote accepted

It's just a conventional choice of words. It means that the set $X$ is a metric space when you think of it together with the metric.

Formally, a metric space is the combination of $X$ with the metric, and if you change the metric you get a different metric space even though $X$ may stay the same.

The conventional way of speech is useful because there are many common cases where there's only one metric under consideration for each $X$ -- then it is convenient to be able to speak about "a subset of the metric space $X$" where for full formal completeness one must otherwise have said "a subset of the underlying set of the metric space $M$", which would be more wordy without really adding any information that's interesting in context.

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It just means that $d$ is a metric on $X$. One could just as well say that a metric space $\langle X,d\rangle$ is a set $X$ together with a metric $d$ on $X$, or that a metric space is an ordered pair whose first component is a set and whose second component is a metric on that set. Of course all of these versions presuppose that the reader already knows what it means for a function $d$ to be a metric on $X$:

  1. $d:X\times X\to\Bbb R$;
  2. $d(x,y)\ge 0$ for all $x,y\in X$, with equality iff $x=y$;
  3. $d(x,y)=d(y,x)$ for all $x,y\in X$; and
  4. $d(x,y)\le d(x,z)+d(z,y)$ for all $x,y,z\in X$.
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To stress Henning Makholm's point, nothing mysterious is happening here. "Meaning is use" (as Wittgenstein almost said). And to grasp the meaning of talk about a set being "equipped" with a function, we just need to grasp how it is used. It isn't used to refer to some novel kind of entity. We are certainly not talking about some sort of hybrid, half set, half function (like an abstract correlate of centaur, half man half horse)! And even to say that we are referring to a pair of a set and function can be misleading, if that is read as implying that we are introducing a new pair-entity, over and above the set and the function: there is no reason to suppose we are doing that either. Talk of a set equipped with e.g. a metric function is just used when we are considering together two things, a set and a function defined over it.

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Thank You! I found your answer to be both entertaining and insightful; and while I thought your question answered my intended question, I chose Makholm's answer as it answered the asked question. – evanjs Oct 8 '12 at 22:06
Good point. I had something like that in mind when I chose to write "combination" rather than "pair". (Now that you mention it, it's surprisingly hard to come up with cases where one needs to treat metric spaces or other sets-with-structure as single first-class mathematical objects. There's category theory, but that's hardly the poster child for the reduce-everything-to-ZFC ideal). – Henning Makholm Oct 8 '12 at 22:15

Metric space is the pair of set $X$ and some function $d\colon X\times X \rightarrow \mathbb{R}$. Function $d$ must satisfy metric axioms. We may define on $X$ different metrics, so obtain different metric spaсes.

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