Is it automatic that a compact Riemann surface $X$ of genus $g>1$ is not homeomorphic to a compact subset of the complex projective line $\mathbb{CP}^1?$
Note: I apologize for the confusion-I am just a high school student with an interest in math. This question was an offspring of my earlier question. Since it seemed sufficiently distant in nature, I decided to ask a new question.
Thanks in advance!
The genus is a topological invariant, in particular a surface of genus $g \geq 1$ can't be homeomorphic to the Riemann sphere. Moreover, any subset of the Riemann sphere has boundary unlike a surface of genus $g$ so such an homeomorphism is also not possible.
On the other hand, any surface $S$ of genus $g \geq 2$ can be obtained as a quotient of the unit disk $D \subset \Bbb C$. A surface of genus 1 is more or less by definition $\Bbb C/ \Lambda$ where $\Lambda$ is a lattice.