Why is the set of all lines through the origin homeomorphic to the circle S1?

  • Thread starter jacobrhcp
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In summary, the problem is to prove that P1, the set of all lines through the origin, is homeomorphic to S1, the circle. The attempt at a solution involves showing that P1 is homeomorphic to S1/R, where R is a set defined by a specific condition. This is done by showing that there is a continuous surjective map from S1 to S1/R, but there is a question about the map being injective. The conversation continues with a discussion about the origin and its placement on the circle, which helps clarify the problem for the individual.
  • #1
jacobrhcp
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Homework Statement



Prove P1 is homeomorphic to S1

The Attempt at a Solution



P1 is the set of all lines through the origin. I have shown for myself that this is homeomorphic to S1/R, where R is given by: (x,y) in R iff x=y or x=-y.
This also shows there is a continuous surjective map f from S1 to S1/R. To prove S1 and S1/R are homeomorphic I need to show that f is injective, and that it has continuous inverse.

However, I think there is something fundamentally wrong with my understanding of the problem, because I cannot imagine the map f being injective. To me it feels as though S1/R is the same as the 'northern hemisphere' of S1, because it sends -x and x to the same line through the origin (where x is an element of the real plane on the circle S1). So wouldn't that imply that f(x)=f(-x) while x is not equal to -x, so f is not injective, so P1 is not homeomorphic to S1?

Anyone who can tell me why I'm fundamentally off here, and give me a push in the right direction, thank you.

Jacob
 
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  • #2
Think of the origin as a point at the top (north pole?) of the circle. As the lines pass through the origin they project on to the circle like a stereographic projection.
 
  • #3
Ah I was taking the centre as origin... it confused me deeply.

But then again, isn't S1={x in the real plane | ||x||=1}... so why do we get to say the origin is on the top of the circle instead of in the middle?

This new view will probably help me take a better shot at the problem later this evening.
 
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1. What is the difference between P1 and S1?

P1 and S1 are both topological spaces, but they have different shapes. P1 is a one-dimensional space that looks like a circle, while S1 is a two-dimensional space that looks like a sphere.

2. How can P1 and S1 be homeomorphic if they have different dimensions?

Homeomorphism is a topological concept that focuses on the shape and structure of a space, rather than its specific dimensions. P1 and S1 have the same fundamental shape, which allows for a continuous mapping between them.

3. What is the significance of P1 being homeomorphic to S1?

The fact that P1 and S1 are homeomorphic means that they are topologically equivalent, and therefore share many of the same properties. This allows for the use of techniques and results from one space to be applied to the other.

4. Can you provide an example of a homeomorphism between P1 and S1?

One example of a homeomorphism between P1 and S1 is the mapping that wraps a line segment around the circumference of a circle. This preserves the shape and structure of both spaces and allows for a continuous transformation between them.

5. Are there other topological spaces that are homeomorphic to P1 and S1?

Yes, there are many other topological spaces that can be homeomorphic to P1 and S1, such as the torus (donut shape) and the figure eight space. Any space with the same fundamental shape and structure as P1 and S1 can be considered homeomorphic to them.

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