*Additional Problem 1:
In this problem you'll prove that S^(n) is the one-point compactification of R^(n). To fix notation:
S^(n)={(x_(1),dots,x_(n+1)):x_(1)^(2)+cdots+x_(n+1)^(2)=1} is the unit sphere in R^(n+1),
view R^(n) as a subspace of R^(n+1),R^(n)={(x_(1),dots,x_(n),0)inR^(n+1)},
and let N=(0,0,dots,0,1) be the north pole of the sphere.
(a) Define f:S^(n)-{N}->R^(n), by letting f(P) be the intersection of the line through N and P with the
hyperplane R^(n). Prove that f is a homeomorphism. (This function is called stereographic projection. You will
probably want to draw a picture in the cases n=1 and 2.)
(b) Prove that f induces a bijection between the set of closed subsets of S^(n) not containing N, and the set of
compact subsets of R^(n).
(c) Now let g:S^(n)->R^(n)cup {infty } have g(N)=infty , and g=f everywhere else. Show that g is a
homeomorphism, where R^(n)cup {infty }=(R^(n))^(+)has the one-point compactification topology.
(d) Here's a case where you'd want a single "point at infinity". Consider the function f(x)=(1)/(x), which is
not defined on all of R. There's a natural way to extend f to an actual function R^(+)->R^(+). Show that this
function is a homeomorphism. If we view R^(+)as the circle, what homeomorphism S^(1)->S^(1) is this?
*Additional Problem 1:
In this problem you'll prove that Sr is the one-point compactification of IR". To fix notation:
. and let N = (0, 0,. . . ,0,1) be the north pole of the sphere.
(a) Define f : Sn -- {N} -> IR", by letting f(P) be the intersection of the line through N and P with the hyperplane IR" . Prove that f is a homeomorphism. (This function is called stereographic projection. You will probably want to draw a picture in the cases n = 1 and 2.)
(b) Prove that f induces a bijection between the set of closed subsets of Sr not containing N, and the set of compact subsets of IRn.
(c) Now let g : Sn > IR U{o} have g(N) = oo, and g = f everywhere else. Show that g is a homeomorphism, where IR" U{o} = (R")+ has the one-point compactification topology.
(d) Here's a case where you'd want a single "point at infinity". Consider the function f() = 1/, which is not defined on all of IR. There's a natural way to extend f to an actual function IR+ -> IR+. Show that this function is a homeomorphism. If we view IR+ as the circle, what homeomorphism S1 --> S1 is this?
