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In the file. 2 questions
PS: d and d' are equivalent if they have exactly the same sequence with the same limit.
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2. Example of two equivalent distances π, π β² on the same set such that (π, π) is totally bounded while (π, πβ² ) is not.
The idea is that convergence is a topological property and it is preserved under a homeomorphism, while boundedness is not in (general).
Let π = β and πβ² be the usual distance. It is obvious that (π, π β² ) is not totally bounded (its length is infinite). Let π be another, βshrunkβ metric:
π π
π(π₯, π¦) = |arctan π₯ β arctan π¦|. The map arctan π₯ actually is a homeomorphism from β with the usual metric to (β 2 , 2 ) with the usual metric.
This function obviously positive except for π₯ = π¦ and symmetric. Prove the triangle...
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