Real projective space
Contents
Definition
Short definition
Real projective space is defined as projective space over the field of real numbers , with its topological structure induced by the structure of the field of real numbers as a topological field.
Finite-dimensional
Real projective space of dimension , denoted or , is defined as the quotient space under the group action where acts by scalar multiplication. It is equipped with the quotient topology.
As a set, we can think of it as the set of lines through the origin in . Using an inner product on , it can also be identified with the set of hyperplanes of codimension 1 (i.e., -dimensional linear subspaces) in .
Countable-dimensional
This space, called countable-dimensional real projective space and denoted , is defined as the quotient space of the nonzero elements of a countable-dimensional real vector space (with the standard topology) over by the action of by scalar multiplication.
Others
We can also consider the real projective space corresponding to any topological real vector space, possibly infinite-dimensional, which is a real vector space equipped with a compatible topology. If the vector space is , the projective space is defined as follows:we take with the subspace topology, and then put the quotient topology on its quotient under the action of .
Particular cases
Real projective space | |
---|---|
0 | one-point space |
1 | real projective line, which turns out to be homeomorphic to the circle |
2 | real projective plane |
3 | link: Fill this in later |
countable () | countable-dimensional real projective space |
Algebraic topology
Homology
Further information: homology of real projective space
Cohomology
Further information: cohomology of real projective space
Homotopy
Further information: homotopy of real projective space