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)})