Problem Set
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Chapter
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Problems
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| 06 |
2 |
22
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∗13 |
Let E be a subset of a metric space X. Show that if E totally bounded then E is bounded. Prove also that, when X = Rk, if E is bounded then E is totally bounded. Give an example of a bounded set which is not totally bounded.
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∗14 |
Suppose that F, K are disjoint subsets of a metric space X, with F closed and K compact. Prove that dist(F, K ) > 0.
Does the result hold if we replace the assumption “K is compact” with “K is closed”?
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∗15 |
Show that the sums of the lengths of intervals removed from the set [0, 1] in order to arrive at the Cantor middle-thirds set is 1.
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{∗16} |
[Note that this one is optional.] Let x and y be numbers in the interval [0, 1] with x < y. Prove that there is a number z with x < z < y such that the ternary expansion of z must have a digit equal to 1.
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