## Supplement to the 4th homework

Posted: Oct 20, 2000

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:

Also show that  R2 - {point} is not the union of two disjoint nonempty open subsets.

Suggested method:

1. Work the "extra" problem.
2. Do a proof by contradiction.
• Specifically, if  R2 - {point} = UV,  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.

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