- 49

- 16

This is not quite what was said. [itex] RP^2 [/itex] (or [itex] CP^2 [/itex]) is a quotient of a sphere so it is already compact. So if you take the one point compactification (which definitely does exist as has already been mentioned since the space is locally compact Hausdorf) you just get [itex] RP^2[/itex] or [itex] CP^2 [/itex] with an isolated point added so this is not interesting at all.however if one is referring to the projective plane R2, as it was pointed out there is not a one point compactification, one needs a circle to compactify it, and it is not homeomorphic to S2 but to its quotient by +/- the identity if I understood it right.

What the others were talking about was compactifying the usual nonprojective plane [itex] \mathbb{R}^2 [/itex] in different ways to get either a sphere or a projective space as a result, not compactifying projective space itself. The example of attaching a circle to the Euclidean plane in a specific way gives you the projective plane which is not homeomorphic to the one point compactification [itex] \mathbb{S}^2[/itex] but it is still a compactification of the plane. How you attach the circle is extremely important though since there are many ways to attach a circle to an open disc (which is the same as attaching it to [itex] \mathbb{R}^2 [/itex] topologically) and the result is usually not the same space. Take a look at the dunce cap for another way of attaching a circle to get something topologically different.

In fact, now that I think about it, it seems every compact connected surface (orientable or not) is the compactification of [itex] \mathbb{R}^2[/itex] since it can be obtained by gluing a circle (usually represented as the boundary of a polygon) to the disc in a particular way and clearly the original disc is dense in the resulting surface.

You need to include the point [itex] 0 [/itex] for the topologists sine curve to be non-path connected. You can't extend the map from [itex] (0,1] [/itex] to the topologists sine curve including [itex] 0 [/itex] in a continuous way.But how can the topologist's sine curve not be a counterexample? It is a continuous surjection

Since you included affine varieties in your previous discussion of embeddings, there are certainly many more embedding type theorems of varieties into projective space. For example, every smooth projective variety of dimension [itex] n [/itex] can be embedded in [itex] \mathbb{P} ^{2n+1}[/itex] (if I recall correctly this requires you to work over an algebraically closed field but I'm not sure if it extends to more general fields or not.)2) According to Wiki, all affine spaces and affine varieties can be embedded in some projective space

In fact the idea of very ample line bundles is precisely one way of giving a variety (the base) enough information (the sections of the sheaf) to embed it into projective space. Of course there is some nonuniqueness to take care of here if you want to set up a correspondence with very ample line bundles and embeddings of varieties into projective space but I seem to remember this being straightforward to deal with. The Kodaira embedding theorem is another such result you may want to look into if interested. I'm sure there are many more but none are coming to mind at the moment.