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:
“...if you are to gain any great amount of good from the world, you must attain a passive condition of mind. ...it is never to be forgotten that it is the rest of the world and not you that holds the great share of the world’s wealth, and that you must allow yourself to be acted upon by the world, if you would become a sharer in the gain of all the ages to your own infinite advantage.”
—Anna C. Brackett (1836–1911)
“nor till the poets among us can be
literalists of
the imagination—above
insolence and triviality and can present
for inspection, ‘imaginary gardens with real toads in them’,
shall we have”
—Marianne Moore (1887–1972)
“With sturdy shoulders, space stands opposing all its weight to nothingness. Where space is, there is being.”
—Friedrich Nietzsche (1844–1900)