Math 121 Homework Assignment #1
Spring 2007
Due Friday April 13
From your textbook:
Page 8 # 7,8,11
Page 12-13 #7
Also:
1. Show that the completion of Q in the sense of problem 7 (on page 12-13) is R.
2. Show that the irrational numbers in R are not a countable union of closed sets.
Deduce that the rational numbers are not a countable intersection of open sets.
3. A condensation point of a set S in a metric space X is (by definition) a point p such
that, for each ε>0, (,)SBp
ε
∩is uncountable (uncountably infinite).
Prove: If S is an uncountable set in R (or more generally in Rn ) then there exists a
condensation point for S.
Suggestion: R=, so there is some interval[n,n+1] such that
[n,n+1]∩S is uncountable. Hence, either [n,n+½]∩S or [n+½,n+1 ] ∩S is
uncountable. Continue to find sequence of “nested” intervals of length
[, 1]nn
∞
−∞
+
∪
1
2k ,
k=1,2,3… each of which has an uncountable intersection with S.