Problem: Let be a compact metric space and beexpansivein the sense that .

Prove:

a)is an isometry.

b)is a homeomorphism.

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- February 24th 2010, 06:27 PMDrexel28Nice result of compactness.
**Problem**: Let be a compact metric space and be*expansive*in the sense that .

Prove:

**a)**is an isometry.

**b)**is a homeomorphism. - February 25th 2010, 07:46 PMBlack
__Spoiler__: - February 25th 2010, 07:55 PMDrexel28
You're first solution is almost identical to mine. The second bit differs slightly.

Assume there exists some , since in any metric space the distance between a point and a compact subspace ( 's continuity implies 's compactness) is positive we have that .

Define, where . This is a sequence in and since every compact metric is sequentially compact there exists some which is convergent, and thus Cauchy. In particular, there exists some such that . But, by assumption . This is of course a contradiction.

Note that you need not have that is an isometry to prove that it's surjective.

There is another way to prove the first one that is a tad more elegant, but a tad more verbose.

Regardless, good job!!