An elementary proof of Freiling's symmetric covering theorem
A collection of closed intervals, C, is said to be a full symmetric cover of \({\mathbb{R}}\) if for every real number x there is a \(\delta (x)>0\) such that \([x-t,x+t]\in C\) whenever \(0<t<\delta (x).\) If C is a full symmetric cover of \({\mathbb{R}}\), C is said to partition an interval [a,b] if there is a partition \(a=u_ 0<u_ 1<...<u_{n+1}=b\) of [a,b] with \([u_ i,u_{i+1}]\in C\) for \(i=0,1,...,n.\) A set \(E\subset {\mathbb{R}}\) is said to be an exceptional set for the cover C if whenever \(x,y\in {\mathbb{R}}-E\) and \(x<y\), then C partitions the interval [x,y]. The authors give an elementary proof of the following theorem of \textit{C. Freiling} [Trans. Am. Math. Soc. 318, No.2, 705-720 (1990; Zbl 0701.26007)]: Let C be a full symmetric cover of \({\mathbb{R}}\). Then (i) there are minimal countable exceptional sets for C; (ii) no minimal countable exceptional set contains a subset which is bilaterally dense in itself.
full symmetric cover of \({\mathbb{R}}\)
scientific article; zbMATH DE number 4210448