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:
“Alas, poor Yorick! I knew him, Horatio: a fellow of infinite jest, of most excellent fancy.... Where be your jibes now, your gambols, your songs, your flashes of merriment that were wont to set the table on a roar?”
—William Shakespeare (1564–1616)
“Children ... seldom have a proper sense of their own tragedy, discounting and keeping hidden the true horrors of their short lives, humbly imagining real calamity to be some prestigious drama of the grown-up world.”
—Shirley Hazzard (b. 1931)
“The merit of those who fill a space in the world’s history, who are borne forward, as it were, by the weight of thousands whom they lead, shed a perfume less sweet than do the sacrifices of private virtue.”
—Ralph Waldo Emerson (1803–1882)