By Jens Peter Reus. Christensen

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X\O Qo UFY . The above reasoning shows t h a t t h e f i r s t category r e l a t i v e t o X\A* . A \ A* i s of But a s e t which i s o f t h e f i r s t category r e l a t i v e t o an open s e t which c o n t a i n s i t i s obviously o f t h e f i r s t category r e l a t i v e t o t h e whole space. We remark t h a t a s e t only i f . i s o f t h e first category. Conversely assume an open set 0 such that C A O Then C* = 0" and hence shows that C * \ C C* that there exist is of the first category.

T h e n a countable i n t e r s e c t i o n o f hence M i s a l s o 6-compact is fi X 6-compact s u b s e t s i n , i s standard. Baire8s Hn/r F i f and o n l y i f . ,M union f of an open A is a as b e f o r e ) . This con- cludes the proof. We denote the coordinatewise ordering in $la by Let 5 . be an arbitrary set equipped with the partial order S . 5 S is (by our definition) an analytically ordered set p:iw-*S if there exist a mapping such that the follo- wing conditions are fulfilled i) n i m ==+ (4 p(n)< ii) For all s 6 S The order relation there exist < .

It is then clear that fa,Bn] is a 32 I H I OKLMS 01 SLPARATION. This concludes the p r o o f . The next theorem is vital for the isomorphism theorems. Then it is easy to see that there exist v and u such that .. By induction we can , f o r all p , THI OK1 M S 01 SLPARATION. LTC 3 cannot be separated by n sets,in particular and A1( (ml,. ,mp)) . A 33 A2( (mi,. Let f:Y -+X and . Now f-’(Si) i=1,2 belongs to S(c&),so: THEOREMS OF SEPARATION, ETC. As can now be used to show that g(Si)6S(g) (1,g)evidently is a sets C1,C2 space,we can find disjoint i=1,2.