The Ham Sandwich theorem and the continuum

In this video we give the Borsuk Ulam theorem: a continuous map from the sphere to the plane takes equal values for some pair of antipodal points. This is then used to prove the Ham Sandwich theorem (you can slice a sandwich with three parts (bread, ham, bread) with a straight planar cut s so that each slice is cut cut in two. Also we give an application to the continuum: the plane is different (not homeomorphic) 3 dimensional space.