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Homework Statement
show that the two topological spaces are homeomorphic.
Homework Equations
Two spaces are homeomorphic if there exists a continuous bijection with a continuous inverse between them
The Attempt at a Solution
I have tried proving that these two spaces are homeomorphic, however I have no idea whether or not I have done it correctly.
I would really appreciate it if somebody could check my work and give me some tips / comment what I could have done better or different
So I began by drawing both spaces
Then I tried to make a "plan" on how I how transform ##x\rightarrow y##
1. rotate ##X## by ##\frac{\pi}{2}##
2. flatten the bottom part of the circle
3. shrink ##X## so that the top of ##X## touches ##Y##
4. finally shrink ##X## to ##y##
1.
##f(r,\theta)=(r,\theta\frac{\pi}{2})## this function is continuous and its inverse ##g(r,\theta)=(r,\theta+\frac{\pi}{2})## is also continuous
2.
##f(x,y)=(x,y+\sqrt{1x^2})## is continuous and its inverse is continuous ##g(x,y)=(x,y\sqrt{1x^2})##
3.
##f(x,y)=(x,y/2)## is continuous and its inverse is continuous ##g(x,y)=(x,2y)##
4.
##f(x,y)=(x,\frac{y(x^4+1)}{\sqrt{1x^2}})## is continuous and its inverse is continuous ##g(x,y)=(x,\frac{y\sqrt{1x^2}}{(x^4+1)})##
as you can probably see I did in 4 steps what my professor would do in 1 maybe 2 functions at most I just find it really hard if I would have tried looking for the function directly.
Also could somebody tell me if this statement is true
the function ##f:\mathbb{R}\rightarrow\mathbb{R}## given with ##f(x)=\arctan{x}## is closed.
Thanks in advance
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