1- let (X,d) metric space and f:X --> X ISOMETRY WITH f[X]is dense in X prove f is homomorphism. 2- suppose (X ,d) metric space has infinitely many distinct connected components. Is it possible (X,d)...

1- let (X,d) metric space and f:X --> X ISOMETRY WITH f[X]is dense in X prove f is homomorphism.

2- suppose (X ,d) metric space has infinitely many distinct connected components. Is it possible (X,d) compact ? if yes ,give an example if no provide proof?

Answered Same DayDec 22, 2021

Answer To: 1- let (X,d) metric space and f:X --> X ISOMETRY WITH f[X]is dense in X prove f is homomorphism. 2-...

Robert answered on Dec 22 2021

130 Votes

1). This is a famous theorem, called completion of a metric space. You
can find the theorem of books on analysis like Goldberg etc. But here is a link
http://www.math.columbia.edu/~nironi/completion.pdf
2) A compact space can have either infinitely many connected components
(eg. Cantor set) or...

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1- let (X,d) metric space and f:X --> X ISOMETRY WITH f[X]is dense in X prove f is homomorphism. 2- suppose (X ,d) metric space has infinitely many distinct connected components. Is it possible (X,d)...

Answer To: 1- let (X,d) metric space and f:X --> X ISOMETRY WITH f[X]is dense in X prove f is homomorphism. 2-...

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