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

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Discussion Overview

The discussion revolves around the topological properties of the sphere minus the north pole and its potential homeomorphism to the plane. Participants explore concepts of compactness, closedness, and the implications of removing a point from a topological space.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant asserts that the sphere minus the north pole cannot be homeomorphic to the plane due to the compactness of the sphere minus a point, contrasting it with the non-compact nature of the plane.
  • Another participant questions the closedness of the sphere minus a pole and the applicability of stereographic projection in this context.
  • A later reply acknowledges the earlier misunderstanding regarding the compactness of the sphere minus a point.
  • One participant reflects on the significant topological changes that occur when a single point is removed from the sphere, noting the effects on compactness and closedness.

Areas of Agreement / Disagreement

Participants express differing views on the properties of the sphere minus a point and its relationship to the plane, indicating that the discussion remains unresolved with multiple competing perspectives.

Contextual Notes

Some assumptions regarding the definitions of compactness and closedness are not explicitly stated, and the implications of stereographic projection are not fully explored.

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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