 By Roman Sikorski

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Extra resources for Boolean Algebras: Reihe: Reelle Funktionen

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1 This theorem is a particular case of a general theorem (due to HEWITT [2J) on Cartesian products of topological spaces. The case m = Xo of this theorem was independently found by MARCZEWSKI . 2 This theorem is a particular case of a more general theorem proved by TARSKI [6J. @2m, m. Then the mapping h(A) = A (\ X for A E 5=Q,2m = is an isomorphism of 5=0. 2m into 5=. G) The Stone space X o of the field5= of all subsets of a set of apower m ;;;; X o has the power 2 2m .! In fact, it is easy to prove that the set of all maximal filters in 5= has a cardinal ~ 2 2m .

1. ) = /1 and consequently ht,(A l ) n ... (A n ) = /1. '. 2. a'ET are independent indexed sets of subalgebras of Boolean algebras Q! ;. o' The following operation of forming products of fields of sets yields an important example of independent indexed set of subalgebras. For every t <: T, let ~t be a (non-degenerate) field of subsets of a nonempty space Xt. Let X be the Cartesian product of an the spaces Xt, i. e. the set of all x = {Xt} = {xthET where xtE X t for every tE T. For every set A C Xt, let A * be the set of all points x E X whose tth co-ordinate X t belongs to A, and let ~i be the field (of subsets of X) composed of all sets A * where A E ~t.

Observe that A ES if and only if h (A) C D, 1. e. h (A) n E = /1. Since the mapping ho defined by the formula ho([A]) = h(A) n E is an isomorphism of Q(/S onto 'JIE, the space E is the Stone space Q(/S. 1 STONE [6J. , and eonversely. Every homomorphie image of Q1. /6, we infer that Stone spaees of homomorphie images of Q1. ' is a homomorphie image of Q1. if and only if its Stone spaee is homeomorphie to a dosed subset of the Stone spaee of Q1.. /6 where 6 is the ideal dual to 5. Thus two elements A, B EQ1.