hw5 - f n[0 1 →[0 1 f n x = the n th digit in the binary...

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Math W4051: Problem Set 5 due Wednesday, October 10 Reading: Munkres, Ch.3 § 26-29, Ch.4 § 30-32, Ch.7 § 43-44. 1. Munkres, Ch.3 § 26, exercise 2 on p. 171. 2. Munkres, Ch.3 § 29, exercise 1 on p. 186. 3. Munkres, Ch.7 § 43, exercise 5 on p. 270. 4. Let X be the real line R in the standard topology, and Y the real line in the trivial topology. Show that X × Y is limit point compact, but neither compact nor sequentially compact. 5. Let X = [0 , 1] [0 , 1] be the product of uncountably many copies of the interval [0 , 1] , indexed over the same interval [0 , 1] . We give X the product topology. Show that X is not sequentially compact. (Hint: Consider the sequence of functions
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Unformatted text preview: f n : [0 , 1] → [0 , 1] , f n ( x ) = the n th digit in the binary expansion of x. ) 6. Prove that [0 , 1] ω ⊂ R ω is not compact in the uniform topology. Prove also that it is not compact in the box topology. (It is compact in the product topology by Tychonoff’s Theorem, but you do not have to show that.) Challenge (extra credit): Let X be a locally compact topological space. Does it follow that every x ∈ X has a neighborhood whose closure is compact?...
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This note was uploaded on 02/10/2010 for the course MATH 205 taught by Professor Smith during the Spring '05 term at Adrian College.

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