- #1

nightingale123

- 25

- 2

## 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{1-x^2})## is continuous and its inverse is continuous ##g(x,y)=(x,y-\sqrt{1-x^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{1-x^2}})## is continuous and its inverse is continuous ##g(x,y)=(x,\frac{y\sqrt{1-x^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