Math 142, Spring 2013. HW8 Solutions
4. Consider a homomorphism f : 1 (S1 , 1) 1 (S 1 , 1); this morphism is completely determined by the value f (1) (since, for n > 0, f (n) = f (1 + 1 + . + 1) = f (1) + . + f (1) and
f (1) = f (1), so f (n) = f (1) . f
Math 142, Spring 2013. HW3 Solutions
3. a) Write
S = Q1 Q2 Q3 Q4 ,
where Q1 is the east quadrant in S, Q2 is the north quadrant in S, Q3 is the west quadrant
in S, Q4 is the south quadrant in S. For example, we have
Q1 = cfw_(x, y ) S | |x| |y |, x 0
and
Math 142, Spring 2013. HW2 Solutions
3. a) Let A X , with X a topological space. Then, A is the intersection of every closed set
that contains A,
A=
F
AF
F closed in X
Thus, as this is an arbitrary intersection of closed sets it must be a closed set (by t
Math 142, Spring 2013. HW7 Solutions
2. a) Denote the constant map with with domain I , R and image cfw_0 I , R, by f0 , by abuse
of notation; hence, f (x) = 0, for each x I , R. Dene the homotopies
H1 : I I I ; (t, x) (1 t)x,
H2 : I R R ; (t, x) (1 t)x.
Math 142, Spring 2013. HW5 Solutions
1. Let A X be connected.
a) If A is closed then A = A so the statement is trivial. Assume that A = A and suppose
that A is not connected. Then, there are nonempty open subsets U, V A such that
U V = and A = U V . Let A
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#1
Curtis Tekell
Prof. Kate Poirier
February 11, 2013
Exercise 1. Consider with its standard metric 0 , (, ) =
.
(a) Prove that the standard metric on is indeed a metric.
(b) Consider with its standard metric. Show that () = ( , + ).
(c) Show th