Show that (0, ∞) is homeomorphic to (0, 1)

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In summary, the author is looking for a function that maps (0, 1) to (0, oo). The function is assumed to be continuous, and the author provides a composition of continuous maps to show that it is. Finally, the inverse of the function is also assumed to be continuous.
  • #1
Mikaelochi
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TL;DR Summary
(1) I need to find a function that maps (0, ∞) to (0, 1) or vice a versa. (2) Show f is a bijection (3) Show that f is continuous (4) that the inverse of f is continuous
So, I already have a function in mind: tan(pi*x - pi/2) that maps (0, 1) to (0, oo). I just forget how to rigorously show that a function is continuous. I was hoping to get some help on showing that this tangent function I just wrote is continuous (not the topological definition, just like the real analysis definition). Rigorously that is. Thanks!
 
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  • #2
Mikaelochi said:
Summary:: (1) I need to find a function that maps (0, oo) to (0, 1) or vice a versa. (2) Show f is a bijection (3) Show that f is continuous (4) that the inverse of f is continuous

So, I already have a function in mind: tan(pi*x - pi/2) that maps (0, 1) to (0, oo). I just forget how to rigorously show that a function is continuous. I was hoping to get some help on showing that this tangent function I just wrote is continuous (not the topological definition, just like the real analysis definition). Rigorously that is. Thanks!
Are you sure that function works?

You mean an epsilon-delta proof that ##\tan## and ##\arctan## are continuous?
 
  • #3
Yeah I mean an epsilon-delta proof that tan and arctan are continuous.
 
  • #4
Mikaelochi said:
Yeah I mean an epsilon-delta proof that tan and arctan are continuous.
I'm tempted to say that with a problem at this level you may assume the continuity of trig functions. Otherwise, you could use some trig identities to crank out a formal proof.
 
  • #5
You have a composition of continuous maps, which is continuous. Can prove this fact in general using epsilon-delta trickery. If you want bijections, there is a good way of getting those via composition.

[tex]
(0,1) \xrightarrow[]{f} (0,\pi/2) \xrightarrow[]{g} (0,\infty)
[/tex]
Put
[tex]
f(x) = \frac{\pi}{2}x \quad\mbox{and}\quad f^{-1}(x) = \frac{2}{\pi}x
[/tex]
also
[tex]
g(x) = \tan x \quad\mbox{and}\quad g^{-1}(x) = \arctan x
[/tex]
Verify you do have inverses i.e ##g\circ g^{-1} ## and ##g^{-1}\circ g ## are the identities. A homeomorphism would then be ##g\circ f ##. As for continuity, note that in ##(0,\infty)##
[tex]|\arctan a - \arctan b|\leqslant |\arctan (a-b)| \leqslant |a-b|[/tex]
Lipschitz maps are continuous.
 
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  • #6
Simpler examples: [tex]
\begin{align*}
x &\mapsto -\ln x \\
x &\mapsto x^{-1} - 1 \\
x &\mapsto x/(1 - x) \\
x &\mapsto \operatorname{arctanh}(x)
\end{align*}
[/tex]
 
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Likes dextercioby
  • #7
Mikaelochi said:
Summary:: (1) I need to find a function that maps (0, oo) to (0, 1) or vice a versa. (2) Show f is a bijection (3) Show that f is continuous (4) that the inverse of f is continuous

So, I already have a function in mind: tan(pi*x - pi/2) that maps (0, 1) to (0, oo). I just forget how to rigorously show that a function is continuous. I was hoping to get some help on showing that this tangent function I just wrote is continuous (not the topological definition, just like the real analysis definition). Rigorously that is. Thanks!
Useful functions to handle with similar question is $$f(x)=\frac{x}{1-|x|}$$ and $$g(x)=\frac{x}{1+|x|},$$ with some change of variable. Note that ##x## can be a vector, like one in ##\mathbb{R}^2##, for instace. Try to search to "\(\frac{x}{1-|x|}\)" on SearchOnMath.
 

1. What does it mean for two sets to be homeomorphic?

Two sets are homeomorphic if there exists a continuous and bijective function between them. This means that the two sets have the same topological structure, even though their elements may be different.

2. How can we show that (0, ∞) and (0, 1) are homeomorphic?

We can show that (0, ∞) and (0, 1) are homeomorphic by finding a continuous and bijective function between the two sets. One possible function is f(x) = 1/x, which maps the interval (0, ∞) to (0, 1).

3. Why is it important to prove that (0, ∞) and (0, 1) are homeomorphic?

Proving that (0, ∞) and (0, 1) are homeomorphic allows us to use the properties of one set to understand the properties of the other set. This can be useful in solving problems or making predictions in various fields, such as mathematics, physics, and engineering.

4. Can any two sets be homeomorphic?

No, not all sets can be homeomorphic. For two sets to be homeomorphic, they must have the same topological structure, which means that they must have the same number of holes, connected components, and other topological properties. For example, a circle and a line segment cannot be homeomorphic as they have different topological properties.

5. How does proving that (0, ∞) and (0, 1) are homeomorphic relate to real numbers?

The real numbers can be represented by a line, where each point on the line corresponds to a unique real number. By showing that (0, ∞) and (0, 1) are homeomorphic, we are essentially showing that these two sets have the same topological structure as the real numbers. This means that they can be thought of as "stretching" or "scaling" the real number line, without changing its topological properties.

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