Proving T(x,y) is a Metric on Compact Set

[Bachelier]

To show that some T(x,y) = something
is a metric on a set for which it is compact, we have to prove that it respects the 3 axioms of distance. right?

 

[Landau]

Is this is a serious question?

 

[HallsofIvy]

Yes, to show that something is a metric, we must show that it satisfies the definition of a metric- which is just those three "axioms" you mention.

In general to show that something is "A" you must show that it satisfies the definition of "A"! I suspect that was the reason for Landau's question.

But I am concerned about that phrase "for which it is compact". The "it" there should refer to the metric you just mentioned but metrics are not "compact". And if you meant the set, whether or not a given function is a metric has nothing to do with whether or not the underlying topological space is compact.