I was reading through the responses to this thread:
$|f(z)|\le 1-|z|\forall z\in D$, we need to show $f\equiv 0$
and I noticed that in accepted answer, that response uses the notation $\partial B(0,r)$ in reference to the disk. I know that this notation comes from the Maximum Modulus Principle, and I was wondering what this partial notation meant in this context. Is this referring to the closure of a set?
In the context of topology, $\partial D$ denotes the (topological) boundary of $D$, i.e. the set of all points $z$ such that any neighborhood of $z$ intersects both $D$ and the complement of $D$.
One version of the maximum modulus principle can then be stated as follows: If $D \subset \Bbb C$ is open and a function $f:\overline D \to \Bbb C$ is holomophic in $D$ and continuous in $\overline D$ then $f$ attains its maximum on $\partial D$.