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:
“Each man has his own vocation. The talent is the call. There is one direction in which all space is open to him. He has faculties silently inviting him thither to endless exertion. He is like a ship in the river; he runs against obstructions on every side but one; on that side all obstruction is taken away, and he sweeps serenely over a deepening channel into an infinite sea.”
—Ralph Waldo Emerson (1803–1882)
“Perhaps the facts most astounding and most real are never communicated by man to man.”
—Henry David Thoreau (1817–1862)
“Shall we now
Contaminate our fingers with base bribes,
And sell the mighty space of our large honors
For so much trash as may be grasped thus?
I had rather be a dog and bay the moon
Than such a Roman.”
—William Shakespeare (1564–1616)