connect5 - 5 Let us conclude these notes with a discussion...

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5 Let us conclude these notes with a discussion of the most disconnected set possible – the Cantor set . Example 20. Let C denote the Cantor set (cf. § .244 in [1]). Recall that C may be described as the set of all even ternary numbers ; that is, C [0 , 1] consists of numbers x whose base-3 expansion [ x ] 3 has only 0 s and 2 s. Let x C , and Fx ± > 0 . Then, by the Archimedean property, there is n N with n ± > 1 . Note that 3 n >n , and so ± > 1 3 n > 0 . Now, write x in base-3, [ x ] 3 =0 .x 1 x 2 x 3 . . . x n - 1 x n x n +1 . . . This is just short-hand for the expansion x = x 1 3 + x 2 3 2 + x 3 3 3 + · · · + x n 3 n + · · · Now, consider the two numbers x + and x - given by x ± = x ± 1 3 n . If x n =0 then [ x + ] 3 =0 .x 1 x 2 . . . x n - 1 1 x n +1 . . . ; if x n =2 then [ x - ] 3 =0 .x 1 x 2 . . . x n - 1 1 x n +1 . . . . In either case, x ± do not have even ternary expansions, and so x ± / C . A similar argu- ment (though with more work due to carrying, as in “carry the
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This note was uploaded on 05/06/2010 for the course MATH Math2009 taught by Professor Koskesh during the Spring '09 term at SUNY Empire State.

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