Show P^1 is homemorphic to S^1

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Homework Help Overview

The discussion revolves around the topic of topology, specifically the homeomorphism between the projective line P^1 and the circle S^1. The original poster seeks assistance in proving this relationship by establishing a suitable function that meets specific criteria.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the definition of P^1 and its relation to lines in R^2. There is mention of using stereographic projection as a potential method for establishing the homeomorphism. Questions about the details of these concepts and their application are raised.

Discussion Status

The conversation is ongoing, with participants exploring definitions and potential approaches. Some guidance has been offered regarding the use of stereographic projection, but there is no explicit consensus or resolution yet.

Contextual Notes

There are uncertainties regarding the precise definition of P^1, particularly whether it refers to lines in R^2 or R^2 excluding the origin. Additionally, the original poster expresses confusion about the details of the proposed methods.

jack8572357
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Show P^1 is homemorphic to S^1
I know I need to prove there is a function satisfying it's 1-1 ,onto,continuous,and the inverse of function is continuous.However, I can't find it.Please help!
 
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What is your definition of [itex]P^1[/itex]?

Knowing about stereographic projection might be helpful as well, so look that up.
 
P^1 is the set of all line in R^2(or R^2\(0,0), I forget which one is right) through the origion.
 
jack8572357 said:
P^1 is the set of all line in R^2(or R^2\(0,0), I forget which one is right) through the origion.

Can you attach to a line in [itex]\mathbb{R}^2[/itex] a real number?? For example, given a line through the origin [itex]ax+by=0[/itex], I can look at the slope [itex]-b/a[/itex] (works if a is nonzero).

So, that gives a function between [itex]P^1[/itex] except one point and [itex]\mathbb{R}[/itex]. Then apply stereographic projection.
 
Would you say it in detail?I have no idea about it.
 

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