Suggestion: Do a proof by contradiction. If [a,b] = U
V,
we may assume that
b 
V.
Define c to be the least upper bound of U.
(Be sure to explain why this exists). Then study what happens in each of
the two cases
c U and
c V.
An "extra" problem
Let a < b. Show that the closed interval
[a,b] is not the union of two disjoint nonempty open subsets
(in the relative topology). (Please note the
correction.)
Suggestion: Do a proof by contradiction. If
[a,b] = UV,
we may assume that
b V.
Define c to be the least upper bound of U.
(Be sure to explain why this exists). Then study what happens in each of
the two cases
c U and
c V.
Scroll way down (or click here)
to see suggestions about exercise 10 on pp. 53-54 or the text.
A further hint about exercise 10 on pp. 53-54
of the text:
Comments and questions to:
roberts@math.umn.edu
Back to the homework list
Back to the class homepage.
Also show that R2 - {point} is not the union of
two disjoint nonempty open subsets.
Suggested method:
V,
show that there are points
A U
and B V
such that the line segment joining A and B
does not contain the point that is being omitted.