Want to know an intuitive but valid way of making metric sets

I just want an intuitive but valid way to describe a set *S* where every element in *S* is a coordinate (*x*, *y*, *z*, *t*) and every component is a real number multiplied by 1 standard unit of measure. This is how I'd like to describe *S*, but I don't know if it's too inconsistent with convention.

Let **R**m = {*x* m | *x* is in **R**, "m" stands for 1 standard meter}. Example, 3.47 m is an element of **R**m.

Let **R**s = {*x* s | *x* is in **R**^{+} or *x* = 0, "s" stands for 1 standard second}. Example, 0.003 s is an element of **R**s.

Define *S* = {(*x*, *y*, *z*, *t*) | (*x*, *y*, *z*, *t*) is in **R**m^{3} $\displaystyle \times$ **R**s}. Example, (2.2 m, 2.2 m, 2.2 m, 5.0 s) is in *S*.

So, answers that would probably help me the most would be like either

"There's no serious inconsistency between that and convention; you're not saying anything terribly confused" or

"The proper way to say what you want, and in keeping with the spirit of being similarly intuitive, is . . ." or

"There's a proper way to say what you want but I'm afraid understanding it involves still more background that you obviously don't have."

Re: Want to know an intuitive but valid way of making metric sets

Quote:

Originally Posted by

**Laurien** "The proper way to say what you want, and in keeping with the spirit of being similarly intuitive, is . . ."

The usual way of saying this is to separate coordinates from units of measure. That is, coordinates are elements of simple ℝ^{4}. Besides, we are given three space vectors not lying in the same plane, a time interval with a convention whether future or past is a positive direction, and a point in space-time from where these vectors should start, i.e., the origin. Then, formally, a coordinate system is the bijection between points in space-time and ℝ^{4} determined by the vectors and the origin.

Re: Want to know an intuitive but valid way of making metric sets

emakarov, that's exactly what I needed to know. Thank you