MAT 708 Spring 2009 Test 1.
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Directions: Make sure to set up and end your proofs appropriately.
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For any closed interval I with positive length, assume that L  I  , C  I  and R  I  form a partition of I into
a left closed interval, a center open interval, and a right closed interval, each with positive length, i.e. there
exists a  b  c  d such that I is the disjoint union of L  I    a, b , C  I    b, c  , and R  I   c, d .
Let I  be a fixed closed interval. If I x1 , x2 ,
L ' s and R ' s , then let I x1 , x2 ,
I x1 , x2 ,
1.
, xn
, xn , L

, xn
 L I x1 , x2 ,
has been defined where x1 , x2 ,
, xn
 and I
x1 , x2 , , xn , R
where each xi varies over L, R and let E 

 R I x1 , x2 ,
, xn
, xn is some finite sequence of
 . Let E be the union of the
n

En .
n 1
Prove that E is uncountable using a Cantor Diagonalization argument. For each infinite sequence

 x1, x2 , x3 ,...
of L ' s and R ' s , feel free to assume
point y x , x
 , but do not assume the
1
2 , x3 ,
n 1
I x1 , x2 ,
, xn
is nonempty and therefore contains a
y x1 , x2 , x3 ,  ’s are different for different infinite sequences
 x1, x2 , x3 ,... . Instead show this as needed.
2.
Prove that E is perfect (a nonempty closed set in which every point of E is an accumulation point of


E ). To make the proof easier, assume lim length I x1 , x2 ,
n
 x1, x2 , x3 ,...
I x1 , x2 ,
3.
, xn
, xn
  0 for each infinite sequence
of L ' s and R ' s . Also feel free to call z a left endpoint if it is the left endpoint of some
and similarly for right endpoint.

Consider the special case in which E is the Cantor Set so that I  0,1 and C I x1 , x2 ,
middle-thirds open interval of I x1 , x2 ,
(a) Suppose y 
xi
3
i
i
, xn
, xn
 is the
.
for some sequence  x1 , x2 , x3 ,... of 0’s and 2’s. Prove that y  E3 . Feel
1
in this special case of the Cantor Set.
27
(b) Let  x1 , x2 , x3 ,...   L, R, L, R, L, R,... and y  I x1 , x2 , , xn for each n . Calculate the rational
free to use that the length of each I x1 , x2 , x3 is
number y and write it as a fraction.
4.
Suppose C is an uncountable closed set. Prove that C contains a perfect subset ( a nonempty closed
set in which every point of E is an accumulation point of E ).