Show the equality of two topologies

Question

given are two sets. S:=$\{z \in C: |z| = 1\} \subset C$ and B:=$\{e^i\phi:\phi \in (\phi_o -\delta,\phi_o + \delta), \phi_o \in R, \delta\gt0\}$ which denotes a Basis for a topology $\mathcal T$. I have to show that $\mathcal T$ = $\{S \cap V:V\in \mathcal O\}$ with $\mathcal O$ being the standard topology in C. I know that i need to show that one set is a subset of the other one and vice versa and have visualized the two sets, but fail to set up an elegant proof. Any help? Thank you very much

Answer 1

Verify that sets of the form $\{re^{i\phi }: a<r<b,\phi_0 -\delta < \phi <\phi_0 +\delta \}$ form a basis for the standard topology of $\mathbb C$. You will then be able to complete the argument.