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So i have two topological spaces on euclidean plane that are defined as:

$X=\{(x,y)\in\mathbb{R^2}|x<0,y<0, x^2+y^2<4\}$

$Y=\{(x,y)\in\mathbb{R^2}|x>1,y<-1\}$

So i have one quarter of ball in the third quadrant and one is just an infinite rectangle if I may call it like that.

So i wonder how can i find a homeomorphism between these two spaces.

Thank you in advance.

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HINT: Think about Argand diagram https://en.wikipedia.org/wiki/Complex_plane, with the ball standing not in the point $(0,0)$ but $(1,-1)$.

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Hint: Start with an easier setting: $X'= \{(x,y):x> 0,y>0, x^2+y^2 < 1 \},$ $Y' = \{(x,y): x> 0, y>0\}.$ The map

$$(x,y) \to \frac{(x,y)}{1-(x^2+y^2)}$$

is then a homeomorphism of $X'$ onto $Y'.$ You're just a few simple maps away from this.

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