Real projective space

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Definition

Short definition

Real projective space is defined as projective space over the field of real numbers \R, 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 n, denoted \R\mathbb{P}^n or \mathbb{P}^n(\R), is defined as the quotient space under the group action \R^{n+1} \setminus \{ 0 \}/\R^* where \R^* 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 \R^{n+1}. Using an inner product on \R^{n+1}, it can also be identified with the set of hyperplanes of codimension 1 (i.e., n-dimensional linear subspaces) in \R^{n+1}.

Countable-dimensional

This space, called countable-dimensional real projective space and denoted \R\mathbb{P}^\infty, is defined as the quotient space of the nonzero elements of a countable-dimensional real vector space (with the standard topology) over \R by the action of \R^* 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 V, the projective space is defined as follows:we take V \setminus \{ 0 \} with the subspace topology, and then put the quotient topology on its quotient under the action of \R^*.

Particular cases

n Real projective space \R\mathbb{P}^n
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 (\infty) 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