S^2 - {N} and R^2 homeomorphic?

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In summary, the conversation discusses the homeomorphism between the sphere minus the north pole and the plane. It is determined that this is not possible because the sphere minus a point is compact while the plane is not. The idea of using the one point compactification of the plane is considered, but it is pointed out that the sphere minus a point is not compact. It is then noted that removing a single point can have a significant topological impact.
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PsychonautQQ
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Is the sphere minus the north pole homeomorphic to the plane? Obviously no, right? Because the sphere minus a point is compact and then plane is not.

I was just thinking that the one point compactification of the plane is homeomorphic to a sphere, because now you have a way to map to the north pole. So I was just thinking, maybe if we subtract the north pole it will be homemorphic to the plane now. What is wrong with my thinking here?
 
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Why is the sphere minus a pole closed? And why shouldn't the stereographic projection work?
 
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Ahh, so the sphere minus a point is not compact, duh. Thanks.
 
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Sort of weird how removing a single point changes so much topologically. Uncountanly-many points in sphere, yet removing a single one changes compactness, closedness ( tho, obviously, not boundedness).
 
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Related to S^2 - {N} and R^2 homeomorphic?

1. What is the difference between S^2 - {N} and R^2 homeomorphic?

Both S^2 - {N} (the two-dimensional sphere with a point removed) and R^2 (the two-dimensional Euclidean space) are topological spaces, meaning they have the same number of dimensions and properties such as continuity and connectedness. However, they are not homeomorphic, which means there is no continuous function that can map one space onto the other without any distortions or breaks. The main difference between them is the presence of the removed point in S^2 - {N}, which creates a hole and changes the overall shape of the space.

2. Can you provide an example of a homeomorphism between S^2 - {N} and R^2?

No, it is not possible to provide a homeomorphism between S^2 - {N} and R^2. As mentioned before, the removed point in S^2 - {N} creates a hole, which cannot be mapped onto the flat surface of R^2 without any distortions. This is a fundamental difference between the two spaces and thus, a homeomorphism does not exist.

3. What does it mean for two spaces to be homeomorphic?

Two spaces are homeomorphic if there exists a continuous function between them that is bijective (one-to-one and onto) and has a continuous inverse function. This means that the two spaces have the same number of dimensions and properties, and can be transformed into each other without any distortions or breaks. In other words, they are topologically equivalent.

4. Are S^2 - {N} and R^2 homeomorphic in higher dimensions?

No, S^2 - {N} and R^2 are not homeomorphic in any dimension. The presence of the removed point in S^2 - {N} creates a fundamental difference between the two spaces, making it impossible to find a continuous function that maps one space onto the other without any distortions. This applies to all dimensions, not just two.

5. How does the concept of homeomorphism relate to topology?

Homeomorphism is a central concept in topology, which is the branch of mathematics that studies the properties of geometric spaces that are preserved under continuous transformations. Homeomorphism allows us to compare and classify different topological spaces based on their fundamental properties, such as dimensions, connectedness, and compactness. It helps us understand the underlying structure of a space and the ways in which it can be transformed without changing its essential properties.

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