I need to undestand a proof in a euclidean space... but my level of math level is of high school. Can you give me some recommendations to understand this proof? This paper is (Rauch, 1978) Am. Math. Month. it resolve the Illumination Problem. I don't understand well that reasons with the boundary, the disc and the closure. Thank you!
Sorry for the English mistakes.
Let's try to figure it out in a Euclidean plane. The boundary is the line around the domain. A disc is a circle center at $P$ with a radius of $r$ (Choose $r$ small enough as in the proof). The closure is the union of the domain and its boundary. What you need to imagine is the definition of compactness. In Euclidean space, compactness is equivalent to closed (i.e. contains its boundary) and bounded. The general definition of compactness is for every open cover, there is always exist a finite subcover. Hope the picture below will help you well. (From https://en.wikipedia.org/wiki/Interior_(topology))