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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