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Metric space has infinitely many distinct connected

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?

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Mathematics: Metric space has infinitely many distinct connected

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