What Is a Metric Space and How Is It Defined?

[sponsoredwalk]

Hi I have two questions,


1.
A metric space is an ordered pair (M,d) where M is a set (which some authors require to be non-empty) and d is a metric on M, that is, a function

$$d : M x M -> R$$


------------From Wikipedia.
http://en.wikipedia.org/wiki/Metric_space#Definition

I just want to give my interpretation of what this says and if I'm reading this wrong could you correct my vocabulary/"mode of thought". :p

This statement is saying that the generalized metric space "d" (say, a Euclidian or Cartesian Plane) maps (joins together in a workable way) sets M to the real numbers R. d - the metric space - is viewed as a function.

2.
Is this the same way the Euclidian R x R plane is viewed when talking of "addition" in the following;

$$+ : (R x R) --> R$$



I would really appreciate it if you could explain where I'm wrong and what is correct.

 

[VeeEight]

"d" is the distance function, or metric, for the metric space. The domain for the distance function are ordered pairs (x,y) where x and y are in M. For example, in the reals, the standard distance function for x, y in R is |x-y|. You can also define different metrics on R, or even on sets of other elements such as functions or vectors, as long as they obey the laws outlined in the wikipedia link.

Things like addition and multiplication are referred to as binary operations.

 

[Mark44]

d is not a metric space - it is a metric that gives the distance between elements in set M. For different metric spaces, the metric d can be different. The ordered pair (M, d) is the metric space.