Real projective space

Definition

Short definition

Real projective space is defined as projective space over the topological field, namely, the field of real numbers.

Finite-dimensional

Real projective space of dimension $n$ , denoted $R P^{n}$ or $P^{n} (R)$ , is defined as the quotient space under the group action $R^{n + 1} ∖ {0} / R^{*}$ 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 $R P^{\infty}$ , 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 $V ∖ {0}$ with the subspace topology, and then put the quotient topology on its quotient under the action of $R^{*}$ .

Particular cases

$n$	Real projective space $R P^{n}$
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