What does discrete subgroup of $ \Bbb{C}$ spans $ \Bbb{C}$ mean ?
Lattice is defined as $ \Bbb{Z}+τ\Bbb{Z}$ for some $τ\in \Bbb{C}- \Bbb{R}$. I heard a proposition which says 'Lattice is discrete subgroup of $ \Bbb{C}$ which spans $ \Bbb{C}$'.
But I'm having trouble to understand what is the definition of 'discrete subgroup of $ \Bbb{C}$ spans $ \Bbb{C}$'.
Thank you for your help.
It means that it spans $\mathbb{C}$ as a real vector space.
(Edit: A discrete subgroup is a subgroup whose subspace topology is discrete. In this case, it's equivalent to saying that around every point there's an open ball containing no other points. The prototypical picture here is the square lattice $\mathbb{Z}[i] \subset \mathbb{C}$.)
It is a theorem, not a definition, that the discrete subgroups of $\mathbb{C}$ with this property are the ones of the form $\mathbb{Z} a + \mathbb{Z} b$ where $a$ and $b$ are not real multiples of each other. Then up to homothety (rotation + scaling) we can assume that $a = 1$ WLOG. Then we get lattices of the form $\mathbb{Z} + \mathbb{Z} \tau$ (where $\tau = \frac{b}{a}$) and the spanning condition is that $\tau$ is not real; this gives us every lattice up to homothety which is why this specific collection of lattices is used in the study of elliptic curves (since homotheties between lattices give complex analytic isomorphisms between the corresponding quotients of $\mathbb{C}$).