Low levels of visibility (Q2730918)

The paper deals with some advanced aspects of interior visibility. A point \(x\in S\subset E^d\) is said to be totally blind if it cannot see via \(S\) any other point of \(S\). (If \(S\) is the closed unit circle in the \(xy\)-plane centered at the origin with an open circular hole of radius \({1\over 2}\) centered at \(({1\over 2},0)\), then the point \((1,0)\in S\) is totally blind.) Two other levels of visibility are introduced and discussed. A point \(x\) in the closure of \(S\) is said to be accessible from \(S\) if there exists another point \(t\in S\) such that the open segment \((tx)\subset S\).NEWLINENEWLINENEWLINEThe paper establishes some simple relations between these concepts. An example is the following.NEWLINENEWLINENEWLINETheorem 2. A point \(x\in S\) is totally blind if and only if \(x\) is not accessible from \(S\).NEWLINENEWLINENEWLINEThe presentation is not the best. Say, in Example 2, by \(M= Q\sim (\bigcup^\infty_{i=1} \text{int }T_i)\), the authors mean \(M= Q\setminus (\bigcup^\infty_{i=1} \text{int }T_i)\). The statement ``\(\text{lnc} M\) is constituted precisely by all the vertices of the triangles'' is inaccurate as the point \(x\), not such a vertex, is also in \(\text{lnc }M\).