I want to show that $n$ dimensional real projection space is compactification of $\Bbb R^n$, how can I do that?
I can show PR^n is compact, and R^n is localy compact, but I have problem to show R^n is dense in RP^n. Is it done by RP^n=R^n+ RP^n-1 ? Or could you suggest me the better way to show, it is dense?
Is there any direct way?
In some note I read this is not true as 1 point compactification, it's true just for $n=1$. here I saw this:https://en.wikipedia.org/wiki/Compactification_%28mathematics%29
Any closed connected manifold is a compactification of $\mathbb{R}^n$.
To show this, we have to find, for each closed manifold $M$, an embedding $\mathbb{R}^n\rightarrow M$. One way to proceed is to use Morse theory. Any closed manifold admits a Morse function with One minimum. The stable manifold of the minimum is an embedded submanifold diffeomorphic to $\mathbb{R}^n$. As there are no other local minima, this must be dense in the manifold.
Of course this uses some theory. I think you are on the right track with your observation. What is an obvious map $\mathbb{R}^n\rightarrow \mathbb{R}P^n\setminus \mathbb{R}P^{n-1}$?