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1 a player throws a fair coin and wins 1 each time its heads but loses 1 each time its tails the player will stop
let an be an increasing sequence of sigma-algebras whose union generates a sigma-algebra s let micro and nu be
1 let x y and z be independent real random variables with eex infin e x infin ey 0 and ez 2 infin show that x ex y
show that for any metric space s if a sub s and x isin a a then there is a bounded continuous real-valued function on a
1 let x d be a locally compact separable metric space show that its one-point compactification is metrizable2 let x be
let k be the tychonoff-c ech compactification of r show that addition from r times r onto r cannot be extended to a
if x t is a locally compact hausdorff space show that x as a subset of its tychonoff-c ech compactification k is
9 let g be the affine group of the line namely the set of all transformations x frarr ax b of r onto itself where a
a uniform space s u is called separated iff for every x y in s there is a u isin u with x y isin u a show that the
1 if x y and z are uniform spaces f is uniformly continuous from x into y and g is uniformly continuous from y into
1 a topological space s t is called perfectly normal iff for every closed set f there is a continuous real function f
1 a show that any open set u in r is a union of countably many disjoint open intervals one or two of which may be
1 let s d be any noncompact metric space show that there exist bounded continuous functions fn on s such that fn x
a c 1 curve is a function t frarr f t gt from r into r2 where the derivatives f 1t and g1t exist and are
1 show that the set rq of irrational numbers with usual topology relative topology from r is topologically complete2
1 show that the closure of a nowhere dense set is nowhere dense2 let s d and v e be two metric spaces on the
let k be a compact hausdorff space and suppose for some k there are k continuous functions f1 fk from k into r such
1 show that for k 2 3 there is a continuous function f k from 0 1 onto the unit cube 0 1k in rk hint let f 2t gt
peano curvesshow that there is a continuous function f from the unit interval 0 1 onto the unit square 0 1 times 0 1
6 prove that each of the following functions f has properties 1 2 and 3 in proposition 243 a f x x1 x b f x
if si di are metric spaces for i isin i where i is a finite set then on the cartesian product s iti isini si let dx
1 if k d is a compact metric space and u isin k show that for any finite m and 0 alpha le 1 f isin lipalpha m f