Description of real projective spaces in various contexts

Question

What I want to know is :

What is the description of real projective spaces (specially $RP^0$, $RP^1$, $RP^2$) respectively in context of topology, geometry and algebra?

I'm searching for simple descriptions included various examples. Any comments or introductory reference would be appreciated.

Answer 1

Let's start with $\Bbb RP^2$ since the rest are analogous.

Description 1: The points of $\Bbb RP^2$ are the 1-dimensional subspaces of $\Bbb R^3$. A line in $\Bbb RP^2$ is a set of such points such that the 1-d subspaces all lie in a single 2-d subpace of $\Bbb R^3$. A point is on a line if the 1-d subspace for the point is contained in the 2-d subspace for the line.

Description 2: (Basically an algebraic translation of the above) Define an equivalence relation on $\Bbb R^3\setminus\{(0,0,0)\}$ by saying $v\sim w$ whenever $v=\lambda w$ for some $\lambda\in \Bbb R$. Then $\Bbb RP^2$ is the set of equivalence classes of this relation $(\Bbb R^3\setminus\{(0,0,0)\})/\sim$.

It would be a good exercise to convince yourself that the equivalence classes are, roughly speaking, "changing lines of $\Bbb R^3$ into points."

From the first description, you can easily verify that all the axioms for a projective plane are satisfied. For instance, "any two distinct lines meet at a point" occurs because the two distinct planes in $\Bbb R^3$ that represent these lines must intersect in a line of $\Bbb R^3$, and that line is a point of $\Bbb RP^2$ on both lines.

Analogously, $\Bbb RP^1$ can be looked at as the set of $1$-d subspaces of $\Bbb R^2$. The entirety of $\Bbb R^2$ translates into a single line in $\Bbb RP^1$, which is the entirety of $\Bbb RP^1$, the projective line.

$\Bbb RP^0$ is the set of $1$-d subspaces of $\Bbb R$, that is, just a single point.

These can be generalized into more general "projective spaces," but this is the basic situation that should drive your intuition.

Answer 2

I agree with every word written in rschwieb's answer, and would like to add another equivalent definition, which can be useful in some topological and geometric contexts.

Let $n$ be any positive integer (the case $n=0$ is not so interesting topologically, and thus omitted here), and let $\mathbb{Z}/2$ act on $S^n$ by the antipodal map. The projective space $\mathbb{R}\mathbb{P}^n$ is then defined to be the space of $\mathbb{Z}/2$-orbits in $S^n$.

Clearly, as a set, this definition is equivalent to the one given by rschwieb. However, it is convenient to use whenever considering $\mathbb{R}\mathbb{P}^n$ as a topological space. Since $\pi:S^n\to\mathbb{R}\mathbb{P}^n$ is a covering map, every point in the projective space has a neighborhood homeomorphic to the Euclidean $n$-space. In fact, $\mathbb{R}\mathbb{P}^n$ inherits also the smooth structure from $S^n$, which makes it a smooth manifold.