Has anyone heard the term before? I am reading a paper by E. Hopf written in 1950 which states the following:
Let $\Omega$ be a bounded open set in $\mathbb{R^n}$. Denote by $p$ a point in $\mathbb{R^n}$. A closed subset of the boundary of $\Omega$ is called accessible from within $\Omega$ if $\Omega$ contains a semi-open Jordan curve $p(t), 0\leq t < \infty$ whose points of accumulation, for $t\rightarrow \infty$, precisely constitute that boundary part.
The language used by Hopf has stumped me. I think a semi-open Jordan curve refers to a curve which gets "arbitrarily close to being closed" but isn't quite. Ie, a curve $p(t)$ which at $p(0)$ is at a point $x$ and as $t$ rushes off to infinity, the curve $p(t)$ keeps getting closer to $x$ but never reaches $x$. However, if someone knows a rigorous definition, it would help a lot.
In context, the obvious guess would be that "semi-open Jordan curve" refers to a continuous injection from a half-open interval, in this case the interval $[0,\infty)$. (Or maybe not just a continuous injection but an embeddding, but that makes no difference since the condition that the accumulation points as $t\to\infty$ are all on the boundary makes it automatically an embedding.)