There is the circle group which is specifically a set of points comprising the boundary of the unit disc in the complex plane which turns out to be a compact set.
However, is the set of points forming a circle of any arbitrary radius also compact in the complex plane?
Yes, either because of Heine-Borel: a closed and bounded (in absolute value ) subset of $\Bbb C$ is compact, or by showing that all maps $f(z)= \alpha z$ are continuous (and have a continuous inverse when $\alpha \neq 0$ in $f^{-1}(z)=\frac{1}{\alpha}z$) and so preserve compactness (the continuous image of a compact set is compact as well).