# Attainable boundary point

A point on the boundary of a domain together with the class of equivalent paths leading from the interior of the domain to that point. Let $\xi$ be a point on the boundary $\partial G$ of a domain $G$ in the complex $z$- plane and let there exist a path described by the equation $z = z (t)$, where the function $z (t)$ is defined and continuous on a certain segment $[ \alpha , \beta ]$, $z (t) \in G$ if $\alpha \leq t < \beta$, $z ( \beta ) = \xi$. One then says that this path leads to the point $\xi$( from the inside of $G$) and defines the attainable boundary point represented by $\xi$. Two paths leading to $\xi$ are said to be equivalent (or, defining the same attainable boundary point) if there exists a third path which also leads to $\xi$ from the inside of $G$ and which has non-empty intersections inside $G$ as close to $\xi$ as one pleases with each of the two paths considered. The totality of a point $\xi \in \partial G$ and the class of equivalent paths leading to $\xi$ from the interior of $G$ is said to be an attainable boundary point of the domain $G$. Not every point $\xi \in \partial G$ represents an attainable boundary point; on the other hand, the same point $\xi \in \partial G$ can represent several, or even an infinite set of different, attainable boundary points.