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:
“Assumptions that racism is more oppressive to black men than black women, then and now ... based on acceptance of patriarchal notions of masculinity.”
—bell hooks (b. c. 1955)
“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 (18711922)
“No being exists or can exist which is not related to space in some way. God is everywhere, created minds are somewhere, and body is in the space that it occupies; and whatever is neither everywhere nor anywhere does not exist. And hence it follows that space is an effect arising from the first existence of being, because when any being is postulated, space is postulated.”
—Isaac Newton (16421727)