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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
let xi i isin i be a net where i is a directed set for j sub i xi i isin j will be called a strict subnet of xi i
18 a real-valued function f on a topological space s is called upper semi- continuous iff for each a isin r f -1a
14 a topological space s t is called connected if s is not the union of two disjoint non-empty open setsa prove
given a product x iti isini xi of topological spaces xi t with product topology and a directed set j a net in x
1 if x t and y u are topological spaces a is a base for t and b is a base for u show that the collection of all sets
1 if si are sets with discrete topologies show that the product topology for finitely many such spaces is also
1 let s be any set and sinfin the set of all sequences xn nge1 with xn isin s for all n let c be a subset of the
8 a let q be the set of rational numbers show that the riemann integral of 1q from 0 to 1 is undefined the net in
1 let x d and y e be pseudometric spaces with topologies td and te metrized by d and e respectively let f be a function
1 on r2 let dx y u v x - u2 y - v212 usual metric ex y x - u y - v show that e is a metric and metrizes the same
1 show assuming ac that any cartesian product of finite sets is either finite or uncountable cant be countably
1 in the partially ordered set n times n with the ordering j k le m n iff j le m and k le n consider the sequence n
1 prove without applying theorem 151 that the well-ordering principle implies ac caution is it clear that the
if x is uncountable and y is a countable subset of x show that xy has the same cardinality as x assuming that n has
data on the magnitudes of earthquakes near fiji are available on the website for this book estimate the cdf f x compute
12 let x x1 x2 x3 be random variables that are positive and integer valued show that xn x if and only if for every