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.
Read more about this topic: Metric Space
Famous quotes containing the words notions of, notions and/or space:
“Your notions of friendship are new to me; I believe every man is born with his quantum, and he cannot give to one without robbing another. I very well know to whom I would give the first place in my friendship, but they are not in the way, I am condemned to another scene, and therefore I distribute it in pennyworths to those about me, and who displease me least, and should do the same to my fellow prisoners if I were condemned to a jail.”
—Jonathan Swift (1667–1745)
“Even the simple act that we call “going to visit a person of our acquaintance” is in part an intellectual act. We fill the physical appearance of the person we see with all the notions we have about him, and in the totality of our impressions about him, these notions play the most important role.”
—Marcel Proust (1871–1922)
“For tribal man space was the uncontrollable mystery. For technological man it is time that occupies the same role.”
—Marshall McLuhan (1911–1980)