Week 6 homework, Math 446.
1. Show that {(x,y) € R² | x>=0 } is not homeomorphic to R². Deduce that the
boundary of a bounded 2-manifold is topologically invariant.
2. Prove that for a polygonal arc p:[0,1]-> R² joining points a and b, there is a subset of
p([0,1]) which is an embedded polygonal arc joining a and b.
3. Show that if a Jordan curve J is not convex, then there is a line L in R² such that
( L intersect J )  is disconnected and one component of R²-L does not meet J. (hint:
take the convex hull of J)