Why is the number of singularities (zeros and poles) of an elliptic function finite in a fundamental domain of a lattice in C?

Question

Let $\Gamma$ be a lattice in $\mathbb{C}$ spanned by $\omega_1$ and $\omega_2$.

1. Denote $\mathbb{C}/\Gamma$ = $\{a\omega_1 + b\omega_2~|~a,b \in [0,1)\}$. Why is $\mathbb{C}/\Gamma$ compact?

2. Suppose $f$ is an elliptic function in $\Gamma$. Why is the number of poles $mod ~\Gamma$ of $f$ finite? (Elliptic function refers to the doubly periodic meromorphic functions.)

3. Let $n_z(f)$ and $n_p(f)$ denote the numbers of zeros and poles of $f$ respectively in $\mathbb{C}/\Gamma$. Why is $n_z(f)$ = $n_p(f)$?