Winter 2014 Math 452
Final
1. (Chapter 14): Let X be a topological space and and be two paths
in X with (0) = (0) and (1) = (1). Prove that if and only
if where is the reverse path of .
Note, on this one I really expect you to get the formula for G.
Proof
Winter 2014 Math 452
Homework One
For Problems 16, let x y be the relation on Z dened by x y if
and only if x y = 3k for some k Z.
1. Show that is an equivalence relationship.
Proof. Reexivity: x x = 0 = 3(0), so x x.
Symmetry: x y means there exists k Z
Winter 2014 Math 452
Homework Two
7. Assume that X Y is path connected and prove that X is path connected. (Hint: this should be a very short proof.)
Proof. The function X : X Y X is continuous and so X (X Y ) =
X is the image of a path connected space by
Winter 2014 Math 452
Homework Three Solutions
All of these are from 25.1(f). I will label the vertices in the polygon with A
being the left-most vertex in the top row of vertices (so on the rst problem,
A is at the tip of arrows a1 and a2 ). From there, v
Winter 2014 Math 452
Midterm Solutions
Work these problems by yourself.
1. Consider the two paths in the complex plane (so 0 = 0 + 0i = (0, 0)
and 1 = 1 + 0i = (1, 0) and consider S 1 = cfw_z C : |z | = 1.
(t) = e2it = cos(2t) + i sin(2t) = (cos(2t), sin(