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(Solved): Problem 1. (i) (Baire Category Theorem) Let (x,p) be a complete metric space, and for each ninN let ...



Problem 1. (i) (Baire Category Theorem) Let (x,p) be a complete metric space, and for each ninN let U_(n)inx be a dense open set. Prove that the set \cap n_(n)=1^(\infty )U_(n)U be any open set, and construct sequences of points x_(n)inx and positive numbers r_(n) such that ()/(bar) (B)_(r_(n))(x_(n))subUn U _(()()n),B_(r_(n+1))(x_(n+1))subB_(r_(n))(x_(n)), and r_(n)->0.G_(j-set) is a subset of a metric space which is a countable intersection of open sets, An ^(()-)F_(\sigma )-set" is one which is a countable union of closed sets. Let (x,p) be a complete and separable metric space with countable dense set D. Prove that D is F_(\alpha ) but not G_(\delta ). (c) Let f:R->R be a function, and let EsubR be the set of points at which f is continuons Prove that E is G_(s)U_(n)={xinR:EE\delta >0AAyinR(|x-y|<\delta Longrightarrow|f(x)-f(y)|<(1)/(n)})


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