Unformatted text preview: interval?), 6,7, 8, 12. Let a 1 , a 2 , a 3 , . . . be an ennumeration of the rationals in [0 , 1]. Choose ² > 0 so that ² < 1 / 8. Consider the union U = [0 , 1] ∩ ∞ [ n =1 ‡ a n² 2 n , a n + ² 2 n · . a) Show that U is an open in [0 , 1], and dense in [0 , 1]. b) Show that [0 , 1]U is not empty. Use the speciﬁc list , 1 , 1 / 2 , 1 / 3 , 2 / 3 , 1 / 4 , 3 / 4 , 1 / 5 , 2 / 5 , 3 / 5 , 4 / 5 , . . . , taking all the numbers with denominator 1, then those with denominator 2, then denominator 3, etc, alwaqys in increasing order, and omitting any numbers that have already appeared in the list. c) Can you ﬁnd an element of [0 , 1]U ?...
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 Fall '07
 HUBBARD
 Math, Topology, Order theory, Metric space, open interval, Closed set

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