Show R^2 \{(0,0)} and {(x,y) | 1 < sqrt(x^2+y^2) < 3} are homeomorphic

[Mikaelochi]

TL;DR Summary

Basically I know that for two things to be homeomorphic the function that maps the two things have to be bijective and continuous but how do I even start this problem?

As I said in the summary, I don't really know how to even figure out which function would be appropriate to map the two sets that I described in the title. I'm using the book called Basic Topology by M.A. Armstrong. The book can sometimes be really dense. I am having a really hard time knowing where to start with this problem because it's not as simple as just proving a particular function is bijective and continuous. I know the first thing is the plane minus the origin and the second thing is a circular disk. Any help or hints or whatever would really help! This course just has me so lost sometimes.

 

[S.G. Janssens]

I would probably begin by showing that $\mathbb{R} \setminus \{0\}$ and $(-3,-1) \cup (1,3)$ are homeomorphic. Next, I would use that the sets in the plane have radial symmetry.

(There are some theorems that can give faster conclusions, but I do not know if you have already proved those. Also, when you are starting out, it may not be bad to actually construct the homeomorphism.)

 

[Mikaelochi]

so for the example you just gave, would I need to find a function that maps R \ {0} to (-3, -1) U (1,3)? I'm not very well versed in homeomorphism proofs

 

[fresh_42]

Summary:: Basically I know that for two things to be homeomorphic the function that maps the two things have to be bijective and continuous but how do I even start this problem?

As I said in the summary, I don't really know how to even figure out which function would be appropriate to map the two sets that I described in the title. I'm using the book called Basic Topology by M.A. Armstrong. The book can sometimes be really dense. I am having a really hard time knowing where to start with this problem because it's not as simple as just proving a particular function is bijective and continuous. I know the first thing is the plane minus the origin and the second thing is a circular disk. Any help or hints or whatever would really help! This course just has me so lost sometimes.

You have a punctured plane and a torus. Draw a circle $\sqrt{x^2+y^2}=2$. This circle will be fixed points. All points $1<\sqrt{x^2+y^2}<2$ will be stretched to $0<\sqrt{x^2+y^2}<2$ and all points $2<\sqrt{x^2+y^2}<3$ will be stretched to $2<\sqrt{x^2+y^2}< \infty .$

Now all you have to do is to figure out the transformations and prove that they are a homeomorphism. Polar coordinates might be easier to handle in this case.

 

[Mark44]

You have a punctured plane and a torus.

Technically, the latter is an annulus. The sets in the problem are in $\mathbb R^2$, not $\mathbb R^3$.

 

[Mark44]

@Mikaelochi , to help you get started with @S.G. Janssens's hint, the tangent function (positive half of the principal branch) maps the interval $(0, \pi/2)$ to the interval $(0, \infty)$. If you can transform the tangent function by compressing it a bit, followed by a translation, you should be able to map the interval $(1, 3)$ to the interval $(0, \infty)$. If you follow that, finding a map from the other half, $(-3, -1)$ shouldn't be too hard.

 

[fresh_42]

I think these approaches with the interval are way too complicated. If we work with rays, we have only one coordinate to bother, the radius. Furthermore, stretchings are linear maps, i.e. determined by two points. This immediately yields continuity and bijection. And if we set $r':=r-2$ we have an origin at $r'=0.$

 

[WWGD]

Summary:: Basically I know that for two things to be homeomorphic the function that maps the two things have to be bijective and continuous but how do I even start this problem?

As I said in the summary, I don't really know how to even figure out which function would be appropriate to map the two sets that I described in the title. I'm using the book called Basic Topology by M.A. Armstrong. The book can sometimes be really dense. I am having a really hard time knowing where to start with this problem because it's not as simple as just proving a particular function is bijective and continuous. I know the first thing is the plane minus the origin and the second thing is a circular disk. Any help or hints or whatever would really help! This course just has me so lost sometimes.

Don't mean to nitpick, but this is an important detail: the homeomorphism is a continuous bijection...with a continuous inverse. Not every continuous bijection has a continuous inverse. I think this is an important detail.

 

[pasmith]

@Mikaelochi , to help you get started with @S.G. Janssens's hint, the tangent function (positive half of the principal branch) maps the interval $(0, \pi/2)$ to the interval $(0, \infty)$. If you can transform the tangent function by compressing it a bit, followed by a translation, you should be able to map the interval $(1, 3)$ to the interval $(0, \infty)$. If you follow that, finding a map from the other half, $(-3, -1)$ shouldn't be too hard.

Alternatively, $\operatorname{arctanh}$ maps $(0,1)$ to $(0,\infty)$.