Extended plane as a topological sphere

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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.
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.

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.

But how can the topologist's sine curve not be a counterexample? It is a continuous surjection
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.

2) According to Wiki, all affine spaces and affine varieties can be embedded in some projective space
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.)

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.
 
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I never said the sphere was added, I said that 1-pt compactifications can be done only on locally-compact Hausdorff spaces by adding a point, and that the result would be a topological sphere by , e.g., the inverse stereographic projection. I guess some types of surfaces can be embedded in complex projective n-space; I don't know the necessary conditions for this to happen. If these surfaces can be embedded in R^n , then I guess we can compose the embedded image can be composed with the inverse stereographic projection for an embedding into S^n. Maybe we can use Whitney embedding theorem : embed a surface in R^4 ( or lower ) , then do the 1-pt. compactification, then the composition of these two would give us an embedding into S^n. Anyone know if there are results for embedding surfaces in CP^n for n>1 ?




Not quite; the geometry of CP^1 is locally-Euclidean, CP^1 being a manifold, but is not Euclidean; for one, CP^1 is compact ( being homeo. to S^2 ) , but Eucliodean space is not.

And the topologists sine curve is an example of a continuous surjection ( onto its image ) from a path-connected space ( the unit interval) , into the non-path-connected topologists sine curve. The OP asked whether path-connected was a topological property and that is what I answered.
That was just obviously a mistake, I was referring to the complex line that is compactified to CP1.
Nowhere in the OP I asked about path-connectedness, my doubt was clarified and it is best for the rest of discussions to start their own thread to avoid further confusion like Terandol interpreting I was trying to compactify the projective plane.
Thanks everyone for your inputs.
 
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I was referring to the complex line that is compactified to CP1.
Sorry to drag this thread out even further than it already has been but this quote gave me the idea that the confusion may have come because of the difference between complex projective space and real projective space and everybody has been misinterpreting everybody else in this thread. I may just be misinterpreting you again and if so just ignore this post but I don't think anybody has explicitly stated that the complex projective line is exactly the same thing as the 2 dimensional sphere in this thread yet which may be what your sources were saying to begin with.

If your book/quotes stated that the one point compactification of [itex] \mathbb{C}^1 [/itex] (or equivalently the real plane) is the complex projective line [itex] \mathbb{C}P^1 [/itex] then this is completely accurate since [itex] \mathbb{C}P^1 \cong S^2 [/itex] so it is orientable. Lavinia didn't say that this wasn't true, but rather he showed you how to compactify [itex] \mathbb{C}^1 [/itex] in a different way by adding a circle to get the real projective plane [itex] \mathbb{R}P^2 [/itex] which is a completely different object than [itex] \mathbb{C}P^1 [/itex] even though they are both 2-dimensional. This is not the one point compactification though and this space is not orientable.
 
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O.K, good, thanks, Terandol, I was just having trouble reconciling two apparently-contradictory statements of the Topologists sine curve and the fact that surjective continuous maps preserve path-connectedness. I guess I could have been more careful reading on the Topologists' sine curve.
 

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