Complex Geometry/Complex manifolds

Definition (complex manifold):

A complex manifold is a manifold of type the category of open subsets of some $\mathbb {C} ^{n}$ with holomorphic maps over [[the site $\operatorname {Ouv} (X)$]], where $X$ is a topological space.

Theorem (holomorphic functions on compact connected complex T1 manifolds are constant):

Let $X$ be a compact connected complex $T_{1}$ manifold and let $\varphi :X\to \mathbb {C}$ be holomorphic. Then $\varphi$ is constant.

Proof: [[Since $X$ is compact and $|\varphi |$ is continuous, $|\varphi |$ attains its maximum on $X$]]. Let $x_{0}\in X$ be the point where it does so. Suppose that $\psi :U\to V\subseteq \mathbb {C} ^{n}$ is a chart st. $x_{0}\in U$. Then $\varphi \circ \psi ^{-1}$ is holomorphic and hence constant by the maximum principle. Thus, we have shown that the nonempty set $\{x\in X||\varphi (x)|=\max _{y\in X}|\varphi (y)|\}$ is open. Since $|\varphi |$ is continuous and $X$ is $T_{1}$, it is also closed. Since $X$ is connected, it therefore equals $X$. $\Box$