homeomorphism proof

**MathSucker** · March 5th 2011, 11:33 AM

I have:

$X= \{ (x,y) \in \Re^2 : 0 < x^2+y^2<1 \}$

$Y= \{ (x,y) \in \Re^2 : 1 < x^2+y^2<2 \}$

both have their usual topologies as subsbaces of $\Re^2$

I want to show that $X \equiv Y$

My work:

Let $S= \{ x:2<x<4 \}$

Let $f$ map $\{x \in \Re^2 : 0 < ||x|| < 1 \}$ to $\{x \in \Re^2: 1< ||x||<2 \}$ by:

$f(x) = (1 + \frac{1}{||x||})x$

Now, $=(1 + \frac{1}{||y||})y$

so $f$ is one-to-one and onto.

$f(f^{-1}(y))=(1+ \frac{1}{||f^{-1}(y)||})f^{-1}(y)$

$=(1+\frac{1}{||f^{-1}(y)||})(1- \frac{1}{||y||})y$

$= ( 1 + \frac{1}{||(1-\frac{1}{||y||})y||})(1-\frac{1}{||y||})y$

$= ( 1 + \frac{1}{(1-\frac{1}{||y||})||y||})(1-\frac{1}{||y||})y$

$= \frac{1}{1-\frac{1}{||y||}}( 1 + \frac{1}{||y||}-\frac{1}{||y||})(1-\frac{1}{||y||})y=y$

and

$f^{-1}(f(x))=(1- \frac{1}{||f(x)||})f^(x)$

$=(1-\frac{1}{||f(x)||})(1+ \frac{1}{||x||})x$

$= ( 1 - \frac{1}{||(1+\frac{1}{||x||})x||})(1+\frac{1}{||x ||})x$

$= ( 1 - \frac{1}{(1+\frac{1}{||x||})||x||})(1+\frac{1}{||x ||})x$

$= \frac{1}{1-\frac{1}{||x||}}( 1 + \frac{1}{||x||}-\frac{1}{||x||})(1+\frac{1}{||x||})x=x$

$f$ and $f^{-1}$ are continuous in the unduced topologies on both sets, so... that's it?

**Drexel28** · March 6th 2011, 10:10 AM

Originally Posted by MathSucker

I have:

$X= \{ (x,y) \in \Re^2 : 0 < x^2+y^2<1 \}$

$Y= \{ (x,y) \in \Re^2 : 1 < x^2+y^2<2 \}$

both have their usual topologies as subsbaces of $\Re^2$

I want to show that $X \equiv Y$

My work:

Let $S= \{ x:2<x<4 \}$

Let $f$ map $\{x \in \Re^2 : 0 < ||x|| < 1 \}$ to $\{x \in \Re^2: 1< ||x||<2 \}$ by:

$f(x) = (1 + \frac{1}{||x||})x$

Now, $=(1 + \frac{1}{||y||})y$

so $f$ is one-to-one and onto.

$f(f^{-1}(y))=(1+ \frac{1}{||f^{-1}(y)||})f^{-1}(y)$

$=(1+\frac{1}{||f^{-1}(y)||})(1- \frac{1}{||y||})y$

$= ( 1 + \frac{1}{||(1-\frac{1}{||y||})y||})(1-\frac{1}{||y||})y$

$= ( 1 + \frac{1}{(1-\frac{1}{||y||})||y||})(1-\frac{1}{||y||})y$

$= \frac{1}{1-\frac{1}{||y||}}( 1 + \frac{1}{||y||}-\frac{1}{||y||})(1-\frac{1}{||y||})y=y$

and

$f^{-1}(f(x))=(1- \frac{1}{||f(x)||})f^(x)$

$=(1-\frac{1}{||f(x)||})(1+ \frac{1}{||x||})x$

$= ( 1 - \frac{1}{||(1+\frac{1}{||x||})x||})(1+\frac{1}{||x ||})x$

$= ( 1 - \frac{1}{(1+\frac{1}{||x||})||x||})(1+\frac{1}{||x ||})x$

$= \frac{1}{1-\frac{1}{||x||}}( 1 + \frac{1}{||x||}-\frac{1}{||x||})(1+\frac{1}{||x||})x=x$

$f$ and $f^{-1}$ are continuous in the unduced topologies on both sets, so... that's it?

I didn't check it 100 %, but I looked at most and it looked correct. Good job!

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