Benz W.'s A Beckman Quarles Type Theorem for Plane Lorentz PDF

By Benz W.

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A ( c o u n t a b l y a d d i t i v e ) 0-1 measure d e f i n e d o n t h e c o l l e c t i o n o f a l l s u b s e t s o f a s e t T which i s n o t c o n c e n t r a t e d a t a p o i n t o f T i s c a l l e d a n U l a m measure. Since t h e e x i s t e n c e o f s u c h a measure i s c l e a r l y a p r o p e r t y of t h e e q u i p o t e n c e c l a s s o f T r a t h e r t h a n j u s t T , t h e c a r d i n a l numbers IT1 f o r which U l a m measures e x i s t are c a l l e d U l a m ( m e a s u r a b l e ) c a r d i n a l s .

Since = {p}, BT being Hausdorff, then T C c(nu(p)) u = cu. UFU (PI If T is Lindelaf (CU) ucu (P) possesses a countable subcover (CUn or, n T = @. Thus p 4 UT. V equivalently, (nun) Though completely regular Hausdorff Lindel6f spaces are replete, the converse is false. To see this, let T denote with the topology generated by the half-open intervals (a,bl. T is completely regular, Hausdorff and LindelEf (see below), hence replete, hence so is TXT. TxT is not Lindel6f however, since the closed subspace {(t,-t) It f TI is discrete and nondenumerable.

To show that xn not (c) not (a), suppose that0 z(x ) = @ so that x is invertible in n n -1 B C(T,R). We show that p UT by showing that (x ) (p) = m. 2-2, for each m € N, B n=l B n=l m there is a net (t ) C n z(x ) converging to p in BT. For E > 0 then by n=l AS p n ' m choosing m sufficiently large we may guarantee that 00 I n ; + l I xn( 5 2-n < E on T. Hence Ix-'(tll) > 1 / ~for each index LI and, by the n=m+l (x-') (p) 1 . The desired result that continuity of (x-') (x-1) 6 (p) = m now follows.

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A Beckman Quarles Type Theorem for Plane Lorentz Transformations by Benz W.


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