Unformatted text preview: N ²,X ( x ) ⊆ ( X \ S ) . 4. (i) Show that a subset of a metric space is bounded if it is totally bounded. (ii) Consider an in nite discrete space (say { x 1 ,x 2 ,x 3 , ···} ). Show that it is bounded but not bounded. ( Hint : consider the discrete metric) (iii) Show that in R n , the bounded subsets are exactly the totally bounded sets. 5. Consider the set of real numbers R and the discrete metric. (i) Show that the discrete metric on R is actually a metric. (ii) What are the subsets of R that are closed in the discrete metric? 1...
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 Fall '09
 Antoine
 Economics, Topology, Metric space, Topological space

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