Existence and Uniqueness
Urysohn proved that an Urysohn universal space exists, and that any two Urysohn universal spaces are isometric. This can be seen as follows. Take, two Urysohn spaces. These are separable, so fix in the respective spaces countable dense subsets . These must be properly infinite, so by a back-and-forth argument, one can step-wise construct partial isometries whose domain (resp. range) contains (resp. ). The union of these maps defines a partial isometry whose domain resp. range are dense in the respective spaces. And such maps extend (uniquely) to isometries, since a Urysohn space is required to be complete.
Read more about this topic: Urysohn Universal Space
Famous quotes containing the words existence and, existence and/or uniqueness:
“In a sense the world dies every time a writer dies, because, if he is any good, he has been a wet nurse to humanity during his entire existence and has held earth close around him, like the little obstetrical toad that goes about with a cluster of eggs attached to his legs.”
—E.B. (Elwyn Brooks)
“The settlement of America had its origins in the unsettlement of Europe. America came into existence when the European was already so distant from the ancient ideas and ways of his birthplace that the whole span of the Atlantic did not widen the gulf.”
—Lewis Mumford (1895–1990)
“Somehow we have been taught to believe that the experiences of girls and women are not important in the study and understanding of human behavior. If we know men, then we know all of humankind. These prevalent cultural attitudes totally deny the uniqueness of the female experience, limiting the development of girls and women and depriving a needy world of the gifts, talents, and resources our daughters have to offer.”
—Jeanne Elium (20th century)