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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
suppose we play a game where we start with c dollars on each play of the game you either double or halve your money
1 computer experiment suppose a coin has probabilitynbsppnbspof falling heads up if we ip the coin many times we would
1 suppose that 30 percent of computer owners use a macintosh 50 percent use windows and 20 percent usenbsp linuxnbsp
1 suppose that a and b are independent events show that ac and bc are independent events2 there are three cards the rst
1 for xed b such that pb gt 0 show that pmiddotb satises the axioms of probability2 you have probably heard it before
1 consider the data in example 86 find the plug-in estimate of the correlation coecient estimate the standard error
100 people are given a standard antibiotic to treat an infection and another 100 are given a new antibiotic in the rst