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I'm taking a basic topology course, and studying for my exam, I'm stuck with an exercise that I'm not sure how to do it. Says the following:

Show that the $ \mathbb{R}^ 2 - {(0,0)} $ plane is a homeomorphism at $ S ^ 1 \times (0, \infty) $

I took the function

$f:\mathbb{R}^ 2 - {(0,0)} \rightarrow S ^ 1 \times (0, \infty) $, $f(rcost,rsent)= ((cost, sent), r)$

But I'm not sure if I defined it correctly, even I do not know how to prove it is a homeomorphism. Could you give me a suggestion? Greetings.

H. Dominguez
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  • The function looks good to me. What is the definition of a homeomorphism? – Kenny Wong Mar 01 '17 at 23:21
  • It does not look good to me. The moment you say "let's describe $\Bbb R^2\setminus{(0,0)}$ via polar coordinates" you are implicitly assuming the thesis. Which means that your work is tautological. You need to write $r$ and $(\cos t,\sin t)$ in terms of $x$ and $y$ explicitly and prove that that function is continuous and bijective. After that, you may want to write its inverse explicitly and prove that it is continuous. –  Mar 01 '17 at 23:24
  • @G.Sassatelli Yes, I see what you mean. – Kenny Wong Mar 01 '17 at 23:28

1 Answers1

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So your defintion has one particular issue: it's not a definition. You actually have to show that there exists a function $f$ such that $f\big(r\cos(t), r\sin(t)\big)=\big((\cos(t), \sin(t)), r\big)$. The problem is that you are supposed to define $f$ in terms of $f(x,y)$ but you have an expression as arguments. So the existance of such function you usually prove by explicite constructions.

In such situations I always try to visualize these transformations. So what is $S^1$? This a sphere, i.e. vectors of length $1$. What is $(0, \infty)$? Positive reals. So positive reals could be used to scale vectors of length $1$ to any length. This intuition gives the following mapping:

$$g:S^1\times (0, \infty)\to\mathbb{R}^2\backslash(0,0)$$ $$g(v, r)=r\cdot v$$

So now how to define the inverse? Well you pick a vector and you have to retrieve two pieces of information from it: its length and its projection onto the sphere (this can be done as long as the vector is nonzero). This gives similar definition:

$$f:\mathbb{R}^2\backslash(0,0)\to S^1\times (0, \infty)$$ $$f(v)=\bigg(\frac{v}{\lVert v\rVert}, \lVert v\rVert\bigg)$$

These functions are continous (being compositions of continous functions: scalar multiplication, inverse of a real, taking norm and taking product of two functions) and are invertible to each other. Thus homeomorphisms.

Side note: you might be surprised but actually my definition of $f$ is exactly the function you were trying to construct. You just have to realize that $\big\lVert\big(r\cos(t), r\sin(t)\big)\big\lVert = r$ for $r>0$.

freakish
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