I am new to Complex Algebra, and I am stuck how to show if a set is connected and open. Let $\phi = \{z: 1 < |z| < 2 $ and $Re(z) > -0.5\}$. I would appreciate any help, but I would love if someone could show detailed explanation.
2026-04-05 19:23:24.1775417004
Show the set is connected and open (Complex Analysis)
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The fact that it is open is a consequence of the fact that it is the intersection of two open sets $$ U_1=f_1^{-1}(]1,2[) \mbox{ with } f_1(z)=|z| \mbox{ continuous } $$ and $$ U_2=f_2^{-1}(]1/2,+\infty[) \mbox{ with } f_2(z)=\Re e(z) \mbox{ continuous } $$ If you draw this $\phi$, you will see how to show that it is arcwise connected (take two points and "pass" on the right).