>>11930966ok, now that i'm reading your post i can answer some of your basic questions.
1. yes, whitney approcimation says we can find an approximation which is arbitrarily close. using the tubular neighborhood theorem and compactness of the embedding of S^n, we find an isotopy of the two objects, which is enough to say the separation is preserved (look at the complement).
2. we get finite number of intersections with the sphere due to the fact that the sphere is compact and that the line is transverse. hence the intersection is a compact zero-manifold which must have finitely many points.
>how nasty is it? i have no idea even how to approach thisit uses signed intersection theory (which is what i'm talking about here) to do some fancy stuff with rays sweeping out paths on the inside between points. look up a proof of the jordan curve theorem for a special case. not hard, just long and messy to formalize.
>For me it seems like anon simply made the whole thing more complicated by intruducing more objects into the picture without actually simplifying anythingif you don't know the tools people use in topology all the time, i.e. intersection theory and transversality, then i don't know why you're so upset about them. it's almost like you just don't like topology.
>>11930975i'm really not talking out of my ass at all
