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
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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.