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It says "In mathematics, a metric space is a set for which distances between all members of the set are defined. Those distances, taken together, are called a metric on the set." on Wikipedia.

I know what a distance function is and the properties it should satisfy. However, I don't understand what metric space notation ($M=(X,d)$) contains or represents. I desperately need a very simple and a clear example like given that $X=\{1,2,3,4,5\}$ and $d$ is the usual metric, $M=(X,d)$ is ...

Thanks.

Larx
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    As you (or Wikipedia) said a metric space is given by the following data 1. A set 2. A distance function on the set. Therefore, the notation $M=(X,d)$ means that $M$ is a metric space and is given by the set $X$ and the metric $d:X\times X\rightarrow\mathbb{R}$. – Yanko Oct 21 '17 at 13:40
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    $M = (X,d)$ just indicates that $M$ is a metric space with underlying set $X$ and metric $d$. This is fairly common notation, and not exclusive to metric spaces. For example, $X = (X,\mathcal{M},\mu)$ can be used to represent a measure space, or $X = (X, \tau)$ a topological space. – Xander Henderson Oct 21 '17 at 13:41
  • Can you give me an example of a particular point in that space based on the X I defined and the usual metric? – Larx Oct 21 '17 at 13:47
  • What is the "usual" metric on a finite collection of points? – Xander Henderson Oct 21 '17 at 13:48
  • $ d(x, y) = |x - y|$, that's what I meant at least. – Larx Oct 21 '17 at 13:51
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    The notation just highlights the fact that a metric space has two components. It's not just a set, it's not just a function. It's the combination of a set with a particular function. Sometimes authors are lazy and just use the underlying set $X$ to mean the metric space, if the distance function is clear from the context. There is nothing deep going on here though, you already understand the notation. – Sam Cassidy Oct 21 '17 at 13:55

2 Answers2

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I think you already understood everything there is to understand.

Just for an easier way of writing we define the metric space.

For example $(\mathbb{R},d)$ where $d(x,y) = \mid x-y \ \mid$ is a metric space but $(\mathbb{R},d_2)$ where $d_2(x,y) = \begin{cases} 0, & \text{if} \ \ x=y \\ 1, & \text{otherwise} \end{cases} \ \ \ \ $is another one that behaves totally differently.

So whenever we talk about a metric space, we also specify which metric we are using (often done implicitly).

justabit
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  • Thank you for your answer. What I am asking is given $d(x, y) = |x - y|$ and a set containing positive integers from 1 to 5, can you write an example point on the metric space $(X,d)$. – Larx Oct 21 '17 at 13:54
  • of course: $X$ defines the set of points. So for $X = {1,2,3,4,5}$, one example of a point is $3$. the distance to another point, let's say $2$ is then $d(3,2) = \mid 3 - 2 \ \mid = 1$. – justabit Oct 21 '17 at 13:57
  • What I understand from this is that $(X,d)$ is a set containing all the possible values that can be generated using the elements of $X$ and $d$- that is ${0,1,2,3,4}$. – Larx Oct 21 '17 at 14:01
  • I'm not sure I understand. $X$ defines all the "points" in your metric space. So if you say we have $ X = {1,2,3,4,5}$, then there are the points $1,2,3,4$ and $5$.

    If on the other hand you define $X = {7,8, \frac{2}{3} }$, then the points are $7,8$ and $\frac{2}{3}$. The distance $d$ just explains how much these points are apart and don't interfere with the number of points.

     – justabit Oct 21 '17 at 14:03
  • So far, so good. What I think and do not understand is that if we say $M=(X,d)$ is a space, then (I might be wrong at this point) we should be able to show particular points in that space. Hence, the question is what are some of those points? Or let me put it in this way, are the coordinates in that space expressed as a tuple where we have a pair from the set and the distance between them or something else? – Larx Oct 21 '17 at 14:09
  • Is it clear? The idea is that everything in $X$ is a point. So every element of the set $X$ is a point. for $X = {7,8, \frac{2}{3} }$ for example $\frac{2}{3}$ is a point – justabit Oct 21 '17 at 14:11
  • yes, it is clear. – Larx Oct 21 '17 at 14:12
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    maybe, let's chat, this is easier this way. But a final remark is that a metric space can not be seen as just a set of points. So you should understand $M = (X,d)$ as the points of $X$ with the relation $d$. Don't try to understand M as a set of points, it doesn't work that way. Did that help? – justabit Oct 21 '17 at 14:13
  • It did help. The way I was trying to perceive the issue was not appropriate in the first hand. Thank you. – Larx Oct 21 '17 at 14:41
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A metric space is a pair of objects: a set $X$, which contains the points in the metric space; and a metric $d$, a function that provides a way of measuring "distances" between points in the set $X$. Because a metric space requires both of these components, we often write them as an ordered pair, say $(X,d)$. Because it is a pain to write $(X,d)$ over and over again, we often reduce this to a single letter, say $M = (X,d)$.

In your example, the points of $M$ are the points of $X$, i.e. the points $\{1,2,3,4,5\}$.

Note that this kind of notation is quite common. We often have objects that consist of many pieces. For example, a topological space consists of an underlying space $X$ and a topology $\tau$. Thus we can write $X = (X,\tau)$ for a topological space. The points of a topological space are the points of $X$. The other piece of the pair ($\tau$) gives some extra information about how the points are related to each other.

Xander Henderson
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