> In article <email@example.com>, > Timothy Murphy <firstname.lastname@example.org> wrote: > >> emwong wrote: >> >> > I am not sure whether I understand you correctly. Does the proof uses >> > the assumption that the map sends parallel lines to parallel lines? >> >> Yes. >> When you say that a map T sends straight lines into straight lines, >> I take it this means that all the points of l are mapped into >> all the points of T(l). >> In that case, if T(l) and T(m) meet in P >> then P must be on l and m, which therefore are not parallel.
> But in spaces of 3 or more dimensions there are non-parallel lines which > do not intersect.
True. I should have said that I was considering the 2-dimensional case. In the construction I gave everything was in one plane.
> -- -- Timothy Murphy e-mail: gayleard /at/ eircom.net School of Mathematics, Trinity College, Dublin 2, Ireland