Charts on the torus as a quotient of $\Bbb R^2$

Question

In

Finding coordinate charts from a torus (square with sides identified) to $\mathbb R^2$.

the answer says we can put charts on the torus by open balls of radius $\frac{1}{2}$ on the boundary of the unit square and sending a point to it's image under the quotient map. Won't $(\frac{1}{2}, \frac{1}{2})$ not be in the image of any chart this way? If we make the radius larger, aren't these maps not injective?