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

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SUMMARY

To prove that T(x,y) is a metric on a compact set, one must demonstrate that it satisfies the three axioms of a metric: non-negativity, identity of indiscernibles, and the triangle inequality. The discussion clarifies that the term "compact" should refer to the set rather than the metric itself, as metrics are not compact. The confusion arises from the phrasing, emphasizing the importance of correctly identifying the elements involved in the proof.

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Bachelier
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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?
 
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Is this is a serious question?
 
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.
 

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