Introduction to the analysis of metric spaces by Giles J.R.

By Giles J.R.

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I / for any x 2 KA and i 2 !. 386. ˛2 ; ˇ2 / if and only if ˛1 < ˛2 and ˇ1 > ˇ2 . Denote by A the family of all subsets of T which are linearly ordered by < (the empty set and the one-point sets are considered to be linearly ordered). Prove that A is an adequate family and X D KA is a strong Eberlein compact space which is not uniform Eberlein compact. 387. (Talagrand’s example) For any distinct s; t 2 ! s; t/ D minfk 2 ! k/g. , let AnSD fA ! s; t/ D ng. X / is K-analytic and hence X is Gul’ko compact) while X is not Eberlein compact.

If s 2 ! "g is finiteg. A/) will be called ˙-products (˙ -products) of real lines.

382. Given an infinite set T suppose that a space Xt ¤ ; is uniform Eberlein compact L for each t 2 T . Prove that the Alexandroff compactification of the space fXt W t 2 T g is also uniform Eberlein compact. 383. Let T be an infinite set. Suppose that A is an adequate family on T . Prove that the space KA is Eberlein compact if and only if TA is -compact. 384. Let T be an infinite set. Suppose that A is an adequate family on T . 1/ \ Ti is finite for every x 2 KA and i 2 !. 385. Let T be an infinite set and A an adequate family on T .

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