# What's the motivation behind metric spaces?

So a metric space is a collection of points together with operations, and where we can determine the distance between any of these points. And it must satisfy 4 axioms which are:

• For all x in that metric space, the distance between x and x is 0.
• For all x,y in that metric space, the distance between x and y is the same as the distance between y and x.
• For all distinct x,y in that metric space, the distance between x and y is strictly positive.
• For any x, y, z in that metric space, the distance between x and z is less than or equal to the distance between x and y + the distance between y and z. (the triangle inequality)

But what is the motivation behind defining such abstract spaces?

• Metric spaces are general spaces where the notion of distance makes sense. One major motivation for studying them are to better understand spaces of functions.
– Seth
Apr 14 '14 at 20:56
• They are general enough that the theory applies in a lot of situations, but specific enough that we can prove many results about them. Apr 14 '14 at 20:58
• I heard that one reason may be to generalize the notion of distance and convergence. There are so many of these spaces, but if you prove something in a metric space it holds for all these spaces. If you prove a limit property in a metric space, it holds for $\mathbb{R}^n$ for any n, if you didn't have the metric space, you would have to prove this for all the n's. The properties I am talking about are the ones where the type of metric, and elements in the metric-space do not matter. Apr 14 '14 at 20:58
• If you consider this abstract, don't look into topological spaces :) Apr 14 '14 at 21:20