Real projective plane $\mathbb RP^2$

The quotient of Sphere which identifies its antipodal points.

This space is homeomorphic to:

• $\left( \mathbb R^3 \setminus \{0\} \right) / \sim$, where $x \sim y$ if and only if $x = c y$ for some nonzero real number $c$.
• The space of $1$-dimensional subspaces of the real vector space $\mathbb{R}^3$, $P(\mathbb{R}^3)$, with the quotient topology induced by the map $\mathbb R^3 \setminus \{0\} \to P(\mathbb{R}^3)$ which sends a vector to its span.
• The quotient of $\left[ -1, 1 \right]^2$ identifying $(t, 1)$ with $(-t, -1)$ and identifying $(1, t)$ with $(-1, -t)$, for each $t$.

Defined in Example 3.51 and Proposition 6.2 of MR 2766102.