Intro to Analysis in-class_Part_23

Intro to Analysis in-class_Part_23 - x by x = d 1 3 + d 2 9...

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3.2 Open sets and closed sets 51 Corollary 3.2.15. The closure of A is the smallest closed set containing A . Definition 3.2.16. B is a dense subset of A iff A B . Example 3.2.5 (The Cantor Set) . Define a nested sequence of sets C k +1 C k by C 0 = [0 , 1] C 1 = [0 , 1 3 ] [ 2 3 , 1] C 2 = [0 , 1 9 ] [ 2 9 , 1 3 ] [ 2 3 , 7 9 ] [ 8 9 , 1] . . . The Cantor set is C := T n =0 C n . Alternative definition: define f 1 ( x ) = x 3 and f 2 ( x ) = x 3 + 2 3 . Then C is the unique nonempty closed and bounded set for which f 1 ( C ) f 2 ( C ) = C . Theorem 3.2.17. 1. The Cantor set is closed. 2. Every point of the Cantor set is a limit point. 3. The Cantor set is totally disconnected, i.e., it contains no open interval. 4. The Cantor set contains uncountably many points. 5. The Cantor set has measure zero (length 0) as seen by 1 - j =1 ( 1 3 ) j . 6. For x [0 , 1] , define the ternary expansion of
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Unformatted text preview: x by x = d 1 3 + d 2 9 + d 3 27 + + d n 3 n + = X k =1 d k 3 k , where d k { , 1 , 2 } . Then the Cantor set consists of exactly those x [0 , 1] which have d k { , 2 } , k . 3.2.3 Exercise: #1,4,7,8,13 Recommended: #2,5,14 #7,13 are short-answer. 1. Suppose U is open, C is closed, and K is compact. (a) Is U \ C open? Is C \ U closed? (b) Is U \ K open? Is C \ K compact? 52 Math 413 Topology of the Real Line (c) If V is open, can U \ V be open? (d) If J is compact, can K \ J be compact? 2. Prove the theorems about the Cantor set, using whichever of the denitions seems best....
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Intro to Analysis in-class_Part_23 - x by x = d 1 3 + d 2 9...

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