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An adherent point of $A$ is a point whose every neighbourhood meets $A$. The set of adherent points is the closure of $A$. This includes all points of $A$, even isolated ones; a limit point in the accumulation-point convention instead requires each neighbourhood to meet $A\setminus \{x\}$.

For random proper simple curves from $0$ to infinity in the complex upper half-plane, the probabilities of avoiding every compact H-hull whose closure excludes zero determine the unparameterized trace law. Every point off a simple chordal curve can be connected to a real boundary interval away from zero within one side of its complement. A countable collection of thin rational polygonal tubes attached to that interval therefore recovers the complement from its avoidance indicators. Joint avoidance of finitely many tubes reduces to avoidance of their filled union. If the union blocks every path from zero to infinity the joint event is empty; otherwise filling pockets does not change the avoidance event. The inclusion-exclusion principle gives all finite indicator distributions. These determine the random complement and hence the trace. Simplicity and properness are hypotheses; the assertion is not a general determination theorem for arbitrary random closed sets.

The boundary of a subset $A$ of a topological space consists of the points whose every neighborhood meets both $A$ and its complement. Equivalently, $\partial A=\overline{A}\setminus A^{\circ }$, where the bar denotes closure and the circle denotes interior.

For $x\in \mathbb{R}$ and $R>0$, the filled half-disc $K=\{z\in \mathbb{H}:|z-x|\le R\}$ is a compact H-hull. Its mapping-out function of a compact H-hull is $g_K(z)=z+R^2/(z-x)$. On the semicircle the image is real, and the inverse branch asymptotic to the identity maps the complex upper half-plane onto the exterior half-disc. Its Laurent series has coefficient $R^2/z$, so $\boxed{\mathrm{hcap}(K)=R^2}$. Using an open half-disc instead requires taking its relative closure first.

The forward orbit closure of $x$ under a continuous map $T$ is the closure of its forward orbit. It is a closed forward-invariant set. If $T$ is invertible, one may instead consider the two-sided orbit closure using $n\in \mathbb{Z}$; the convention should be specified. For a minimal point of an invertible compact system the two closures agree.