# Whether or not riemann surfaces can be singletons (and consequences and generalisation)

I refer to Chapter II.3 of Rick Miranda - Algebraic curves and Riemann surfaces, which I understand says in Proposition II.3.9 that injective holomorphic maps $$F: X \to Y$$ between Riemann surfaces $$X$$ and $$Y$$, both of which are connected but not necessarily compact, are isomorphisms onto their images $$F(X)$$. (Connected is part of the definition of Riemann surfaces in this book. Not sure about other books.)

Question 1: Must (the underlying set of) a Riemann surface necessarily not be a singleton? I kind of assume this for the next questions 2 and 3.

Question 2: How is $$F(X)$$ necessarily a Riemann surface?

This is what I tried:

1. $$X$$ is connected and so since $$F$$ is continuous (since holomorphic implies continuous), $$F(X)$$ is connected (since connectedness is preserved under continuity).

2. $$F$$ is not constant: If $$F$$ were constant, then $$F(X)$$ is a singleton. Since $$F$$ is injective, we have that $$X$$ is a singleton if $$F(X)$$ is a singleton. I conclude $$F$$ is not constant if Riemann surfaces cannot be singletons (see Question 1).

3. $$X$$ is of course open in itself, and so since $$F$$ is open (by Proposition II.3.8, which applies, assuming (2)), we have that $$F(X)$$ is open.

4. $$F(X)$$ is Hausdorff and 2nd countable since these properties are inherited from $$Y$$.

5. Finally, for the atlas, Miranda has a recipe for open connected subsets of Riemann surfaces to be Riemann surfaces.

Question 3: If we now suppose $$X$$ is compact, then is $$F$$ surjective?

This is what I tried:

1. Proposition II.3.11 says that a non-constant holomorphic map with a compact domain is surjective has its image/range as compact.

2. If Riemann surfaces cannot be singletons (see Question 1), then I think $$F$$ injective implies $$F$$ non-constant and therefore, Proposition II.3.11 is applicable.

Question 4: To possibly strengthen Question 1, must (the underlying sets of) Riemann surfaces be either uncountable or empty since Riemann surfaces are locally holomorphic/ homeomorphic/ diffeomorphic to open subsets of the complex plane $$\mathbb C$$?

• If $X=\{x\}$ then $\{x\}$ is the only chart around $x$, but there's no homeomorphism $\{x\}\to D$ where $D$ is the open unit disk in $\mathbb C$. Sep 5 '20 at 8:08
• @RandyMarsh Right, that's what I thought. Thanks. Follow-up 1. post as answer for question 1? i'll upvote even if you don't answer the others. Follow-up 2. So this extends to non-empty countable i.e. Question 4 is answered affirmatively? Follow-up 3. Is that riemann surfaces are not singletons indeed relevant for answering questions 2 and 3 affirmatively? Sep 5 '20 at 8:11
• Also, $X=\{x\}$ is compact, however the "smallest" compact Riemann surface is the Riemann sphere. Sep 5 '20 at 8:11
• @RandyMarsh ummmm....ok thanks...i guess smallest is in the sense that riemann sphere is holomorphically embedded or something in every compact riemann surface? Sep 5 '20 at 8:46
• Smallest in the sense of (topological) genus. It is the unique compact Riemann surface with genus $0$. Sep 5 '20 at 9:43

1. Yes. If $$X=\{x\}$$ then $$\{x\}$$ is the only chart around $$x$$, but there's no homeomorphism $$\{x\}\to D$$ where $$D$$ is the open unit disk in $$\mathbb C$$.
2. In general $$F(X)$$ will not be a Riemann surface. However, if $$F$$ is injective, then $$F(X)$$ will be: Let $$w\in F(X)$$, let $$w=F(z)$$ and $$(U,\phi\colon U\to V)$$ a chart around $$z\in X$$. Then $$F(U)$$ is an open neighbourhood of $$w$$ in $$F(X)$$ and the map $$\phi\circ F^{-1}\colon F(U)\to V$$ is a homeomorphism, hence a chart for $$F(X)$$. This, of course, fails when $$F$$ is not injective. The compatibility of charts is immediate from $$(\phi_1\circ F^{-1})\circ(\phi_2\circ F^{-1})^{-1}=\phi_1\circ\phi_2^{-1}$$ (when defined).
4. They have exactly the cardinality of $$\mathbb C$$. They must be uncountable and thus can't be empty. This is a general fact about connected Hausdorff manifolds, see e.g. https://mathoverflow.net/questions/67962/cardinality-of-connected-manifolds