Topology: homeomorphism between quotient spaces

In summary, the conversation discusses showing that four quotient spaces, R^2, R^2/D^2, R^2/I, and R^2/A, are homeomorphic. The poster first shows that R^2 is homeomorphic to R^2/D^2 and then seeks to prove that R^2 is also homeomorphic to R^2/I. They provide a complicated proof and ask for feedback and suggestions for a simpler method. They also propose using a similar method to show that R^2/A is homeomorphic to R^2/I. The conversation ends with a technical question about the homeomorphism of quotient spaces and a request for feedback on their proof.
  • #1
jjou
64
0
I posted this earlier and thought I solved it using a certain definition, which now I think is wrong, so I'm posting this again:

Show that the quotient spaces [tex]R^2, R^2/D^2, R^2/I,[/tex] and [tex]R^2/A[/tex] are homeomorphic where [tex]D^2[/tex] is the closed ball of radius 1, centered at the origin. [tex]I[/tex] is the closed interval [tex][0,1]\in\mathbb{R}[/tex]. [tex]A[/tex] is a union of line segments with a common endpoint (without loss of generality, we can assume the common endpoint is the origin and the first line segment is the interval [0,1] on the x-axis).

I showed that [tex]R^2[/tex] ~ [tex]R^2/D^2[/tex].

Showing [tex]R^2[/tex] ~ [tex]R^2/I[/tex]: We define [tex]R^2/I[/tex] by the equivalence relation [tex](x_1,0)[/tex] ~ [tex](x_2,0)[/tex] iff [tex]x_1,x_2\in[0,1][/tex]. I think it is enough to show that [tex]R^2/[-1,1][/tex] ~ [tex]R^2/D^2[/tex] since changing the interval shouldn't make a drastic difference (I could just rescale & recenter the original segment [tex]I[/tex]).

My proof of this is quite complicated (I think), so would somebody mind checking it / suggesting a more elegant proof? My proof is as follows:

It is enough to find a continuous, surjective map [tex]f:R^2/D^2\rightarrow R^2/[-1,1][/tex] to show the two spaces are homeomorphic. I define this function [tex]f[/tex] to send each point in [tex]D^2[/tex] to its projection on [-1,1]: [tex]f(r,\theta)=(r\cos\theta,0)[/tex] for [tex]r\leq1[/tex].

Then, for points outside of [tex]D^2[/tex], I consider the function [tex]r(\theta)=e^{\theta}[/tex] for [tex]0\leq\theta\leq\pi/2[/tex]. This defines a portion of a spiral-like curve starting at the point (1,0). I reflect this curve over the y-axis for [tex]\pi/2<\theta\leq\pi[/tex] and then reflect over the x-axis for [tex]\pi<\theta<2\pi[/tex].

For points outside of the "spiral," I define [tex]f(r,\theta)=(r\theta)[/tex] (it is the identity map). For points [tex](r,\theta)[/tex] where [tex]1<\theta\leq e^{\theta}[/tex] (i.e. for points outside of D^2 but within the spiral), I in essence "stretch" the segment [tex]((1,\theta),(e^{theta},\theta)][/tex] to cover [tex]((0,\theta),(e^{\theta},\theta)][/tex]. (The actual formulation of this stretching is a bit convoluted, but I have it.)

Then this map [tex]f[/tex] is clearly surjective. Furthermore, it is sequentially continuous (we only need to check the boundaries [tex]r=1[/tex] and [tex]r=e^{\theta}[/tex]). The only "fishy" points are [tex](1,0)[/tex] and [tex](\pi,0)[/tex], but it should work out.

Can anyone find anything wrong with this proof or suggest a simpler method? (I'm almost certain the solution should not be this complicated!)

PS: I think, to show that [tex]R^2/A[/tex] ~ [tex]R^2/I[/tex] can be done by induction and through a method similar to the one described above but with even more reflections - thus, an extremely ugly process.
 
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  • #2
Technical question:
If a set A is homeomorphic to another set B, is it true that the quotient space A/S is homeomorphic to the quotient space B/S (assuming the partition S makes sense for both sets A and B)?

Would really appreciate some feedback on this & the question above. Thanks. :)
 

1. What is topology?

Topology is a branch of mathematics that studies the properties of geometric objects that are preserved under continuous transformations, such as stretching, bending, and twisting.

2. What is a homeomorphism?

A homeomorphism is a bijective function between two topological spaces that preserves their topological structures. In other words, a homeomorphism is a continuous and invertible mapping between two spaces that preserves the relationships between their elements.

3. What are quotient spaces?

Quotient spaces are a type of topological space that are obtained by identifying and collapsing certain points in a given space. This process creates a new space with different topological properties.

4. How do you determine if two quotient spaces are homeomorphic?

To determine if two quotient spaces are homeomorphic, you can first check if there exists a continuous and bijective function between them. Then, you can show that this function is also invertible and preserves the topological structure of the spaces.

5. What is the significance of homeomorphisms between quotient spaces?

Homeomorphisms between quotient spaces allow us to understand and compare the topological properties of different spaces. They also help us to study the relationships between different spaces that may appear different on the surface but have similar underlying structures.

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