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By Benz W.

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N, such that ind Xi = 0 for each i. Let U be a finite open cover of X and let i be such that 0 5 i 5 n. Then the open cover { U n Xi : U E U } of Xi can be refined by an open cover Vi = { V,, : Q E Ai } of order 1. 7 there is an open collection { W;, : Q E Ai } of order 1 in X such that Wi, n Xi = Via. For each Q in A; we select a U i , in U such that 6, c U i , n X i . It is readily verified that the collection V: = { W;, n U i , : Q E Ai } has order 1 and refines U. Thus the open cover V * = V: U .

8 of SklX. The definition slightly differs from the original one of Sklyarenko. 8. We have chosen our definition for two reasons. 8 is used. 8 avoids a pitfall that arises from the other definition. 12. Consider the following example. 13. Example. Let X be the subset of R 2 defined by x = { (5,Y) : z # 0 1u { (070) >. We begin with the following three open sets: UO = { ( z 7Y) : (5,Y) E X and Y > 0 >, u l = s l ( ( - 1 , 0 ) ) = { ( z , y ) : ( ~ t + ) 2 + Y 2< I } , u 2 = Sl((2’0)) = { (5,y) : (5 - 2)2 + y2 < 1).

For every separable metrizable space X , cmp X = n if and only if def X = n. Related to the conjecture of de Groot are the following questions that have motivated this book, 1. What are necessary and sufficient internal conditions on separable metrizable spaces X so that defX 5 n? 2. Is it possible to obtain a fruitful generalization of dimension theory by replacing the empty space in the definition with other spaces? 3. What is the special role of the empty space in the theory of dimension? 5 .

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A Beckman-Quarles type theorem for finite desarguesian planes by Benz W.


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