325 HW 05 - MAT 325: Topology Professor Zoltan Szabo...

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Unformatted text preview: MAT 325: Topology Professor Zoltan Szabo Problem Set 5 Rik Sengupta rsengupt@princeton.edu March 6, 2010 1. Munkres, p. 178, problem 4 Show that a connected metric space having more than one point is uncountable. Solution. Let X be a connected metric space having more than one point. Let x and y be two distinct points of X . Set D = d ( x,y ) > 0. Let f : X R be the continuous function defined by f ( z ) = d ( x,z ) (it is continuous because of the trivial argument with metric continuity between the two spaces). Since X is connected, f ( X ) is connected, too. Hence [0 ,D ] f ( X ). But [0 ,D ] is uncountable, because there is a bijection between this set and the set [0 , 1] (given simply by f ( x ) = x D ), and this last set is clearly uncountable from Cantors uncountable, because (0 , 1) [0 , 1]. Thus f ( X ) is uncountable. Hence X must be uncountable, too. 2. Munkres, p. 178, problem 6 Let A be the closed interval [0 , 1] in R . Let A 1 be the set obtained from A by deleting its middle third ( 1 3 , 2 3 ). Let A 2 be the set obtained from A 1 by deleting its middle thirds ( 1 9 , 2 9 ) and ( 7 9 , 8 9 ). In general, defined A n by the equation A n = A n- 1- [ k =0 1 + 3 k 3 n , 2 + 3 k 3 n . The intersection C = \ n Z + A n is called the Cantor set ; it is a subspace of [0 , 1]. (a) Show that C is totally disconnected. (b) Show that C is compact. (c) Show that each set A n is a union of finitely many disjoint closed intervals of length 1 / 3 n ; and show that the end points of these intervals lie in C ....
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325 HW 05 - MAT 325: Topology Professor Zoltan Szabo...

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