Homework 06 - , 1 } be a function. Show that X is connected...

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MATH 74 HOMEWORK 6 Due Friday March 13th at 3:10pm. Throughout, let X and Y be metric spaces. (1) Let A and B be sets and f a function. Show that f - 1 ( A T B ) = f - 1 A T f - 1 B and that f - 1 ( A S B ) = f - 1 A S f - 1 B . (2) In a general metric space, do open and closed balls have to be connected? (3) Show that the only connected subsets of Q are the empty set and single points. (Sets for which this is true are called totally disconnected. ) (4) Show that being connected is a homeomorphism invariant, i.e. show that if X and Y are homeo- morphic, then X is connected iﬀ Y is connected. (5) Show that R 2 -{ x } is connected but R -{ x } is not. Deduce that R and R 2 are not homeomorphic. (6) Show that if a set contains a nonempty proper subset which is both open and closed then it is disconnected. (7) Let X be a metric space and f : X → {
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Unformatted text preview: , 1 } be a function. Show that X is connected if and only if the only continuous f are constant functions. (8) A space X is called path connected if for every x,y X , there is a continuous function : [0 , 1] X with (0) = x and (1) = y ( is a path or curve connecting x and y ). Show that if a space is path connected, then it is connected. (9) If we lay out an accurate map of California on the oor of our classroom, prove that exactly one point on the map lies directly over the place in California it represents. (10) Let x > 0 and x n +1 = 1 1+ x n . Use the xed point theorem to prove that the sequence { x n } converges and then nd its limit. 1...
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This note was uploaded on 05/04/2011 for the course MATH 73 taught by Professor Danberwick during the Spring '09 term at University of California, Berkeley.

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