Non-Trivial Metric Spaces: Infinite Points by Convention?

[pivoxa15]

Is it by convention that all non trivial metric spaces have an infinite number of points?

Just like all non trivial sequences has an infinite number of points.

 

[morphism]

What do you mean by "non-trivial"?

 

[pivoxa15]

Good one. Probably what I mean is the examples that appear in any textbook. So all examples of sequences in textbooks have them as infinite sequences. Do all examples of metric spaces have infinite number of points?

 

[Crosson]

Do all examples of metric spaces have infinite number of points?

Hmm. In algebra it is frequently used that polynomial spaces (sometimes finite) are metric spaces under a 'difference of degrees' type relation. No one calls it a metric space, but the three defining properties are invoked frequently.

But really my answer is yes, I have only seen metric space terminology in connection with topology and analysis, so non-trivial => infinite...

 

[HallsofIvy]

A finite set with the "discrete metric": d(u,v)= 0 if u= v, 1 otherwise is a finite metric space. Is that what you would consider "trivial"? I wouldn't.

 

[pivoxa15]

Does a finite set with the discrete metric have any applications or anything else of importance?

A set like Q with the discrete metric is non trivial but there are an infinite number of points in it.