Topological spaces

Let $q: [0, 1]\to[0, 1]/(0\sim1)$ and $\phi: [0, 1]\to S^1$, $t\mapsto e^{2\pi it}$. (a) Show $\phi$ induces a continuous bijection $\bar\phi: [0, 1]/{\sim}\to S^1$. (b) Show $\bar\phi$ maps open sets to open sets near the glued point by checking the image of $q([0, \varepsilon)\cup(1 - \varepsilon, 1])$. (c) Conclude $\bar\phi$ is a homeomorphism.