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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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) .
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

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

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online