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18.100B.Homework2

# 18.100B.Homework2 - 6 Probelm 30 page 46 in Rudin 7(a Show...

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18.100B/C: Fall 2008 Homework 2 Available Monday, September 15 Due Wednesday, September 24 Turn in the homework by 11am on Wednesday, September 24, in 2-108. For 18.100B it should be put in the bin corresponding to the lecture you regularly attend (regardless of which one you may be officially enrolled in). If you are enrolled in 18.100C put your homework in the C bin in any case. 1. Let E and F be two compact subsets of the real numbers R with the standard (Euclid- ian) metric d ( x, y ) = | x - y | . Show that the Cartesian product E × F = { ( x, y ) : x E and y F } is a compact subset of R 2 with the metric d 2 such that if v , u R 2 then d 2 ( u, v ) = u - v 2 . Recall that the norm · 2 is defined so that if v = ( x, y ) R 2 , then v 2 = ( x 2 + y 2 ) 1 / 2 . 2. Consider the notes titled Compactness vs. Sequentially Compactness posted on the web page. Prove Lemma 3 stated on those notes. 3. Problem # 12 page 44 in Rudin . 4. Probelm # 14 page 44 in Rudin . 5. Probelm # 16 page 44 in Rudin
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Unformatted text preview: . 6. Probelm # 30 page 46 in Rudin . 7. (a) Show that the Cantor set C = T n ∈ N E n (deﬁned in Rudin 2.44) is uncountable. [ Hint : Use ternary numbers and a diagonal argument similar to Rudin 2.14 .] (b) Show that, for any ± > , there is a union of intervals with total length < ± that contains the Cantor set C . [ Hint : C ⊂ E n , and each of the 2 n intervals in E n is contained in an open interval of length (1 + ± ) / 3 n ]. (c) Show that the Cantor set C ⊂ R is compact. 8. Assume ( X, d ) is a connected metric space. Prove that the only subsets that are both open and closed are X and ∅ . 9. If in a metric space ( X, d ) we have B ⊂ A ⊂ X , then the set B is a connected subset of ( A, d ) (i.e. A with the relative topology) if and only if B is connected subset of ( X, d ) ....
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