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MIT18_100BF10_pset3

# MIT18_100BF10_pset3 - < ± that contains ∩ the Cantor...

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18.100B : Fall 2010 : Section R2 Homework 3 Due Tuesday, September 28, 1pm Reading: Tue Sept.21 : relative topology, compact sets, Rudin 2.28-35 Thu Sept.23 : compact sets, Rudin 2.36-44 1 . Let E and F be two compact subsets of the real numbers R with the standard (Euclidian) metric d ( x, y ) = | x y | . Show that the Cartesian product E × F = { ( x, y ) | x E and y F } is a compact subset of R 2 with the metric d 2 ( �u,�v ) = �u �v 2 . (Recall that the norm � · � 2 is defined by ( x, y ) 2 = ( x 2 + y 2 ) 1 / 2 . ) 2 . Problem # 12 page 44 in Rudin . 3 . Problem # 14 page 44 in Rudin . 4 . Problem # 16 page 44 in Rudin . 5 . Problem # 30 page 46 in Rudin . 6 . (a) Show that, for any � > 0 , there is a union of intervals with total length < �

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Unformatted text preview: < ± that contains ∩ the Cantor set C = n N E n (deﬁned in Rudin 2.44). [ Hint : C ² E n , and each of the 2 n intervals in E n is contained in an open interval of length (1 + ± ) / 3 n ]. (b) Show that the Cantor set C ² R is compact. (not for credit) Show that the Cantor set is uncountable – either by ﬁxing the proof of Rudin 2.43, or by using another (e.g. diagonal) argument. MIT OpenCourseWare http://ocw.mit.edu 18.100B Analysis I Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms ....
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