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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

Tom
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    One-point compactifications are unique up to homeomorphism, so this works only when $S^n$ is homeomorphic to $\mathbb{R}P^n$, which happens only for n = 1. – Pedro Apr 21 '16 at 09:02
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    Adding to Pedro's comment: for $n>1$ we have $\pi_1RP^n=\mathbb Z/2 \neq 0 = \pi_1S^n$, hence there cannot even be a homotopy equivalence. – Daniel Valenzuela Apr 21 '16 at 09:10
  • @Pedro, @ Daniel. in which condition we have this uniqueness up to homeomorphism? always in 1 point compactification? – Tom Apr 21 '16 at 09:27
  • In the real projective space you add an infinite in each direction, so obviously more than a point for $n>1$. – Martín-Blas Pérez Pinilla Apr 21 '16 at 10:00
  • @Martin, Could you say a little explicitly why is it? for example for dim 1, why PR is R + {infinity}? – Tom Apr 21 '16 at 10:55
  • In dim 1 there is only one direction. See https://en.wikipedia.org/wiki/Homogeneous_coordinates, https://en.wikipedia.org/wiki/Point_at_infinity. – Martín-Blas Pérez Pinilla Apr 21 '16 at 11:41

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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 $\mathbb{R}^n\rightarrow \mathbb{R}P^n\setminus \mathbb{R}P^{n-1}$?

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Thomas Rot
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