Is the Projective Real Plane Compact?

Question

I feel like $\Bbb P (\Bbb R^2)$ is compact, but I know that $\Bbb R^2$ is locally compact, therefore it has a one-point compactification. $\Bbb P (\Bbb R^2)$ adds more than one point to the real plane, so I am a little confused. Not sure how to think about this problem.

Answer 1

If you're using the usual topological description, then the short answer is yes: We can describe the projective plane as the quotient of $S^2$ (the sphere) by the action of $\{\pm 1\} \cong \mathbb{Z}/2\mathbb{Z}$. As the quotient of a compact space it is automatically compact (continuous image of a compact set).

As a comment, there isn't only one way to compactify a space. One point is minimal, Stone-Cech is in a certain sense maximal, and there are quite often a few in between.

Answer 2

The one-point compactification of $\mathbb{R}^2$ is a quotient of $\mathbb{P}^2(\mathbb{R})$, that is, smaller than $\mathbb{P}^2(\mathbb{R})$. To get $\mathbb{P}^2(\mathbb{R})$ you add to $\mathbb{R}^2$ the line at infinity which is in fact a projective line, $\mathbb{P}^1(\mathbb{R})$ ( homeomorphic to a circle).

As a side note, there are many compactifications, and the one-point compactification is, for locally compact and non compact, the smallest one. For compact spaces the smallest compactification is the space itself.