(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider the following subsets of [tex]\mathbb{C}[/tex], whose

descriptions are given in polar coordinates. (Take [tex]r \geq 0[/tex] in

this question.)

[tex]

\begin{align*}

X_1 =& \{ (r,\theta) | r = 1 \} \\

X_2 =& \{ (r,\theta) | r < 1 \} \\

X_3 =& \{ (r,\theta) | 0 < \theta < \pi, r > 0 \} \\

X_4 =& \{ (r,\theta) | r = \cos 2\theta \}

\end{align*}

[/tex]

Give each set the usual topology inherited from [tex]\mathcal{C}[/tex].

Which, if any, of these sets are homeomorphic?

2. Relevant equations

3. The attempt at a solution

[tex]\tau_1 = \varnothing[/tex]. [tex]\tau_2 = \{ B(z,r') \cap X_2 | r'

> 0 \}[/tex]. [tex]\tau_3 = \{ B(z,r') \cap X_3 | r' > 0 \}[/tex]. [tex]\tau_4 =

\varnothing[/tex].

[tex]X_2[/tex] is homeomorphic.

Are my answers correct? I am not sure if the topologies I wrote make sense at all.

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# Homework Help: Finding topologies of sets in complex space

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