Infinite Real Projective Space
The infinite real projective space is constructed as the direct limit or union of the finite projective spaces:
Topologically, this space is double-covered by the infinite sphere, which is contractible. The infinite projective space is therefore the Eilenberg-MacLane space and it is BO(1), the classifying space for line bundles. More generally, the infinite Grassmannians are the classifying spaces for finite rank vector bundles.
Its cohomology ring modulo 2 is
where is the first Stiefel–Whitney class: it is the free -algebra on, which has degree 1.
Read more about this topic: Real Projective Space
Famous quotes containing the words infinite, real and/or space:
“Nothing has shown more fully the prodigious ignorance of human ideas and their littleness, than the discovery of [Sir William] Herschell, that what used to be called the Milky Way is a portion of perhaps an infinite multitude of worlds!”
—Horace Walpole (1717–1797)
“O.J. Berman: Well, answer the question now. Is she or isn’t she?
Paul: Is she or isn’t she what?
O.J. Berman: A phony.
Paul. I don’t know. I don’t think so.
O.J. Berman: You don’t think so, huh? Well, you’re wrong. She is. But on the other hand, you’re right. Because she’s a real phony.”
—George Axelrod (b. 1922)
“Through space the universe encompasses and swallows me up like an atom; through thought I comprehend the world.”
—Blaise Pascal (1623–1662)