Metric space notation (X,d) 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.
 A: 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).
A: 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.
