Given a set $X$ in the Euclidean plane $\mathbb{R}^2$ and a point $p \in X$, we say $p$ is accessible if there exists an arc $A \subset \mathbb{R}^2$ such that $A \cap X = {p}$. This is an old and vital notion in plane topology and complex analysis, dating back to Schoenflies in the early 1900’s.

For a given planar continuum $X$, in different embeddings of $X$ in $\mathbb{R}^2$, the set of points of $X$ which are made accessible may vary. One may ask, then, for a given point $p \in X$, does there exist an embedding $\varphi$ of $X$ into $\mathbb{R}^2$ for which $\varphi(p)$ is accessible, or is there some topological obstruction in $X$ which forces $p$ to be inaccessible in every embedding?

In 1972, Nadler and Quinn asked a question in this spirit: For any arc-like continuum $X$, and any point $p \in X$, does there exist an embedding $\varphi$ of $X$ into $\mathbb{R}^2$ for which $\varphi(p)$ is accessible? I will discuss some background for this problem, and describe our recent work in which we give an affirmative answer. This is joint work with Andrea Ammerlaan and Ana Anusic.