What's the numebr of connected components for the set of complex numbers $\{e^z:|z|=1\}$ on the complex plane?
Remark: It represents a simple closed curve which intersects the real axis at points $(\frac{1}{e},0)$ and $(e,0)$, so I think the number of connected components is 1.
Notice that the function $f(z)= e^z$ is continuous and the fact that $A=\{z||z|=1 \}$ is connected. Therefore $f(A)=\{e^z | |z|=1\}$ is connected.