This appendix collects a few basic notions about subsets of \(\R^n\) that appear throughout the book. The point is not to develop general topology, but to fix the language we will use.
Let \(A \subseteq \R^n\text{.}\) A point \(\va \in \R^n\) is a cluster point (or accumulation point) of \(A\) if every open ball centered at \(\va\) contains a point of \(A\) different from \(\va\text{.}\) Equivalently, for every \(r \gt 0\) there exists \(\vx \in A\) such that \(0 \lt \norm{\vx-\va} \lt r\text{.}\)
Let \(A \subseteq \R^n\text{.}\) A point \(\va \in A\) is an interior point of \(A\) if there exists \(r \gt 0\) such that the open ball \(\{\vx \in \R^n : \norm{\vx-\va} \lt r\}\) is contained in \(A\text{.}\)
Let \(A \subseteq \R^n\text{.}\) A path in \(A\) from \(\va \in A\) to \(\vb \in A\) is a continuous map \(\gamma \colon [0,1] \to A\) such that \(\gamma(0)=\va\) and \(\gamma(1)=\vb\text{.}\) The set \(A\) is path connected if every two points of \(A\) can be joined by a path in \(A\text{.}\)
A subset \(A \subseteq \R^n\) is star-shaped if there exists a point \(\vp \in A\) such that for every \(\vx \in A\) and every \(t \in [0,1]\text{,}\) the point
\begin{equation*}
(1-t)\vp+t\vx
\end{equation*}
lies in \(A\text{.}\) We then say that \(A\) is star-shaped with respect to \(\vp\text{.}\)
Fix \(\vp \in A\text{.}\) If \(\vx \in A\) and \(t \in [0,1]\text{,}\) then convexity gives \((1-t)\vp+t\vx \in A\text{.}\) So \(A\) is star-shaped with respect to \(\vp\text{.}\)
Each piece traces a line segment lying in \(A\text{,}\) so \(\gamma([0,1]) \subseteq A\text{.}\) Also \(\gamma(0)=\va\) and \(\gamma(1)=\vb\text{,}\) so \(\gamma\) is a path in \(A\) from \(\va\) to \(\vb\text{.}\) The final sentence follows from Proposition A.0.8.
