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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 i there is an > 0 such that N ( 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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This note was uploaded on 05/20/2010 for the course ECON 511 taught by Professor Johnnachbar during the Fall '08 term at Washington University in St. Louis.
- Fall '08