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Let E = {x + iy : x = 0, or x > 0, y =...

Let E = {x + iy : x = 0, or x > 0, y = sin(1/x)}. Prove that the set E is connected, but that is not path-connected or connected by trajectories

and consider B = {z ∈ C : |z| = 1}. prove ( is easy to see it but how to prove it ) the following questions ¿is B open?¿is B closed?¿connected?¿compact?

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