In measure theory, the order topology on $\overline{\mathbb{R}}$ (extended real) and $[0,\infty]$ provides rich foundation to analyze measurable functions and abstract integral.
Just like this, i heard that there is a topological setting which makes complex analysis fluent, namely Riemann Sphere.
I have proved that the one point compactification $\mathbb{C}\cup\{\infty\}$ of $\mathbb{C}$ and 2-dimensional sphere $S^2$ are homeomorphic.
I heard that in complex analysis, $\frac{1}{0}$ is defined to be $\infty$ and it is quite useful. THe problem is, the text using in the class does not appeal this point of view. Since i'm quite comfortable with topology and measure theory, i want to study complex analysis with a deep point of view.
What else useful topological settings are there to study complex analysis?
Moreover, is there a text written with this aspect?