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Unformatted text preview: x * , there is an open segment of the line that (a) contains x * and (b) is a subset of A . The length of the segment can depend on the line. The property of being internal is related to, but distinct from, the property of being interior . Recall that if ( X,d ) is a metric space and A ⊆ X , then x * is interior to A iﬀ there is an ε > 0 such that N ε ( x ) ⊆ A . (a) Provide an example in R 2 of a set A and a point x * such that x * is internal to A but not interior to A . A picture with a clear accompanying explanation is ideal. (b) Prove that if x * is interior to A ⊆ R N then it is internal to A . (c) Consider ( R ∞ ,d p ), where d p is the pointwise convergence metric. Is the point (1 , 1 ,... ) internal to R ∞ + ? If yes, provide a proof; if no, provide a concrete, clear counterexample....
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 Fall '08
 JohnNachbar
 Economics, Topology, Metric space, Limit of a sequence, Euclidean space, Compact space

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