Editing Real projective space (section)

===Low-dimensional examples===
* {{tmath|\mathbb{RP}^1}} is called the [[real projective line]], which is [[topology|topologically]] equivalent to a [[circle]]. Thinking of points of {{tmath|\mathbb{RP}^1}} as unit-norm complex numbers <math>z</math> up to sign, the diffeomorphism {{tmath|\mathbb{RP}^1 \to S^1}} is given by <math>z \mapsto z^2</math>. Geometrically, a line in <math>\mathbb{R}^2</math> is parameterized by an angle <math>\theta \in [0, \pi]</math> and the endpoints of this closed interval correspond to the same line.
* {{tmath|\mathbb{RP}^2}} is called the [[real projective plane]]. This space cannot be [[Embedding|embedded]] in {{tmath|\mathbb{R}^3}}. It can however be embedded in {{tmath|\mathbb{R}^4}} and can be [[Immersion (mathematics)|immersed]] in {{tmath|\mathbb{R}^3}} (see [[Boy's surface|here]]). The questions of embeddability and immersibility for projective {{tmath|n}}-space have been well-studied.<ref>See the table of Don Davis for a bibliography and list of results.</ref>
* {{tmath|\mathbb{RP}^3}} is [[diffeomorphic]] to [[SO(3)]], hence admits a group structure; the covering map {{tmath|S^3\to\mathbb{RP}^3}} is a map of groups Spin(3) → SO(3), where [[Spin group|Spin(3)]] is a [[Lie group]] that is the [[universal cover]] of SO(3).