connect3 - 3 Example 12. In Example 8, we saw that C1 = [0,...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
3 Example 12. In Example 8, we saw that C 1 = [0 , 1] in the metric space [0 , 1] (2 , 3) . Fol- lowing Lemma 9, this means that for any point x [0 , 1] , C x = [0 , 1] (indeed, this is easy to show directly). On the other hand, if y (2 , 3) , the reader can readily verify that C y = (2 , 3) . Hence, the metric space [0 , 1] (2 , 3) has two connected components: [0 , 1] and (2 , 3) . Example 13. Consider the metric space Q equipped with the usual metric d ( q,p )= | q - p | . Let q 0 be any non-zero rational number. By the Archimedean property of Q , there exists a natural number n so that n · q 2 0 > 1 . Since n 2 +1 >n , we therefore have ( n 2 + 1) · q 2 0 > 1 . Now, de±ne A = { p Q ; p 2 > 1 1+ n 2 } and B = { p Q ; p 2 < 1 1+ n 2 } . By de±nition, q 0 A . Note, 1+ n 2 is not a perfect square (since n> 0 here), and therefore (following standard proofs like the one in the ±rst lecture) 1 1+ n 2 is not the square of any rational number. It follows that Q = A
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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.

Ask a homework question - tutors are online