Note: I'm learning complex analysis from Gamelin and Riemann surfaces appear there in the first chapter as prerequisites. I have no background in Topology/Complex Analysis so please don't use that while answering this.
I don't understand what is actually a Riemann surface (because it's definition involves terms which I don't know), but based on intuitive explanation given in Gamelin I get the following notion of a Riemann surface (I don't know how much is this correct and this may be very wrong):
Let $f(z)$ be a multivalued complex function.
Let $S \in \mathbb{C}$ be the set of points such that for any "sufficiently" small loop $\gamma$ near a point $P \in S$ it's not possible to define $f$ continiously on $\gamma$.
Now we make a bunch of slits $s_1, \cdots, s_k$ (ray/line segment) in $\mathbb{C}$ such that each point $P \in S$ is adjacent to atleast one slit. Now define $f$ continuously on $T:= \mathbb{C} - \{s_1 \cup s_2 \cup \cdots s_k \}$ as $f_1$. Now consider all such functions $f_1, \cdots, f_n$ and glue the slitted complex plane's ($T$) together to form a surface $S$ such that the value of $f$ on $S$ is continuous.
This $S$ is the required Riemann surface
It's not clear to me why the surface $S$ obtained should be independ of in which way you make the slits or how you define $f$ over the slitted complex plane and glue them up.
Consider $f(z) = \sqrt{z(1-z)}$. Here $S = \{0, 1 \}$. Now I think there're two ways to make a slit: either slit $[0,1]$ or slit both $(-\infty, 0]$ and $[1, \infty)$. Doesn't which way you make the slit changes the resulting Riemann surface ?
Also suppose you make the only slit $[0,1]$. Now consider two small circles $\gamma_0, \gamma_1$ centered at $0$ and $1$ respectively. When you approach the slit $[0,1]$ through a variable point $p \in \gamma_0$ in the counterclockwise direction, the sign of $Re(p)$ either is postive or negative. Similar for $\gamma_2$, so total four ways (I guess) you can define $f$ continuously on $\mathbb{C}-[0,1]$ - i.e w. Among them, why it's not possible to define $f$ on $\mathbb{C} - [0,1]$ so that it's like the either of the cases in the right side ?
