Unformatted text preview: #4. The boundary of a subset E of a metric space X , ∂E := { p ∈ X : ∀ r > both E ∩ N r ( p ) and E c ∩ N r ( p ) are nonempty } , where N r ( p ) := { q ∈ X : d ( p,q ) < r } . Show that ∂E is closed in X . #5 (exercise 18 on p.44). Is there a nonempty perfect set in R 1 which contains no rational number? Hint. This problem is not so easy. You can try to adjust the construction of the Cantor set in 2.44 on p.41....
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 Fall '09
 Topology, Sets, 1 K, Metric space, nonempty open subset, arbitrary sequence E1

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