Is S^2 - {N} Homeomorphic to R^2?

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SUMMARY

The discussion centers on the topological properties of the sphere minus the north pole (S^2 - {N}) and its relationship to the plane (R^2). It is established that S^2 - {N} is not homeomorphic to R^2 due to the compactness of the sphere minus a point, which contrasts with the non-compact nature of the plane. The conversation highlights the implications of removing a single point from a compact space, emphasizing how this action alters topological characteristics such as compactness and closedness.

PREREQUISITES
  • Understanding of basic topology concepts, including homeomorphism and compactness.
  • Familiarity with the properties of spheres and planes in a topological context.
  • Knowledge of stereographic projection and its implications in topology.
  • Concept of one-point compactification and its effects on topological spaces.
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  • Research the concept of compactness in topology and its significance in homeomorphisms.
  • Study the properties of one-point compactification and its applications in various topological spaces.
  • Explore stereographic projection in detail, including its role in mapping between the sphere and the plane.
  • Examine examples of other topological transformations and their effects on compactness and closedness.
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Mathematicians, topology students, and anyone interested in understanding the nuances of topological spaces and their properties.

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.
 
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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