hmw4s

# hmw4s - 18.100B, Fall 2002, Homework 4, Solutions Was due...

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18.100B, Fall 2002, Homework 4, Solutions Was due by Noon, Tuesday October 1. Rudin: (1) Chapter 2, Problem 22 Let Q k R k be the subset of points with rational coeﬃcients. This is countable, as the Cartesian product of a ﬁnite number of countable sets. Suppose that x =( x 1 ,...,x k ) R k . By the density of the rationals in the real numbers, given ±> 0 there exists y i Q k such that | x i y i | <±/k , i =1 ,...,k. Thus if y =( y 1 ,y 2 ,...,y k )then k | x y |≤ k max i =1 | x i y i | shows the density of Q k in R k . Thus R k is separable. (2) Chapter 2, Problem 23 Given a separable metric space X, let Y X be a countable dense subset. The product A = Y × Q is countable. Let { U a } ,a A, be the collection of open balls with center from Y and rational radius. If V X is open then for each point p V there exists r> 0 such that B ( p, r ) V. By the density of Q in X there exists y Q such that p B ( y, r/ 2) .

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## This note was uploaded on 12/16/2011 for the course STAT 5446 taught by Professor Frade during the Fall '09 term at FSU.

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hmw4s - 18.100B, Fall 2002, Homework 4, Solutions Was due...

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