How can a metric space be expressed as a set of variables?

Question

I saw a post on here along the following lines:

Consider a metric space $(S,D,K)$.

edit: The original post said $(X,d)$; not sure if this makes a significant difference.

When a metric space is described as a set of variables, does this just indicate the dimensions of the metric space as well as the ordered set of coordinates which the points in this space will have? For instance, for a metric space $(X,Y)$, is it just a two-dimensional space in which every point has a coordinate $(X,Y)$ in which $X$ and $Y$ are determined with respect to $(0,0)$? Or, for any $n$-dimensional metric space, the origin $(0_1,0_2...0_n)$?

Here is the post for reference:

Here is the link: https://mathoverflow.net/q/312176/128941

Answer 1

It could be Product Metric if S, D, K do not share the same metric. So the metric of the set $(S,D,K)$ is,

$d(x,y) = ||(d_S(x_S,y_S), d_D(x_D,y_D), d_K(x_K,y_K))||$

I've read a text in the past, which described a union of sets using that notation. I guess it will depend on whether a metric is given to you.