Let K be a closed subset of the closed unit ball in R3. Suppose there exists a family of chords Ω of the unit sphere S2, with the following property:
for every X,Y∈S2, there exist X′,Y′∈S2, as close to X and Y correspondingly, as we want, such that X′Y′∈Ω and X′Y′ is disjoint from K.
Verify that there exists a set H⊂S2, such that H is dense in the unit sphere S2, and the chords connecting any two points of H are disjoint from K.EDIT: The statement fixed. See post #4 real analysistopologycollege contests