Metric Space - Notions of Metric Space Equivalence

Notions of Metric Space Equivalence

Given two metric spaces (M1, d1) and (M2, d2):

  • They are called homeomorphic (topologically isomorphic) if there exists a homeomorphism between them (i.e., a bijection continuous in both directions).
  • They are called uniformic (uniformly isomorphic) if there exists a uniform isomorphism between them (i.e., a bijection uniformly continuous in both directions).
  • They are called isometric if there exists a bijective isometry between them. In this case, the two metric spaces are essentially identical.
  • They are called quasi-isometric if there exists a quasi-isometry between them.

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