My topology textbook says the following:
Let $S\subset \Bbb{R^2}$ be a closed disc. Then every point in $S$ is contained in a neighbourhood which is homeomorphic to that portion of a ball in $\Bbb{R^2}$ where $x_1,x_2\geq 0$.
I don't see how this is true for a point on the boundary of $S$. Any help would be great.
Any point on the boundary will have a neighbourhood homeomorphic to a neighbourhood in $\mathbb{R}^2_+$; think of this pictorally - its neighbourhood will be a collection of interior points of the disk and a segment of the boundary of the disk. The same is true of a neighbourhood of $\mathbb{R}^2_+$ lying on its boundary.
