## Axiomatic Set Theory: Impredicative Theories of Classes by R. Chuaqui

By R. Chuaqui

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S i m i l a r l y f o r unions. 6 and 5 Z} C_X = F ( X CZ) . 5. * 2 . 3 . 2 . 8 E X A M P L E , In a topological space X, l e t F ( A ) be t h e c l a s s o f accumulation points of A f o r A C X . Let Z be a closed subset of X. T h u s , we have A C 8 C Z + F ( A ) CF(i3) Z. 4, t h e r e i s a l a r g e s t D such t h a t D = F ( D ) . Then D i s p e r f e c t and Z % D i s s c a t t e r e d (Theorem o f Cantor-Bendixon. ) From t h e theorems proved, we now deduce theorems f o r two unary operations.

E. ROLAND0 C H U A Q U I 38 P R O O F OF (v). So (*) R*(A U 8 ) C - (R*A) U (R*B) w i l l be shown. L e t us suppose t h a t ((R*A) u ( R * B ) ) n C = 0. Then, R * A n C = O = R * B n C . By (i), we o b t a i n A n R - l * C = O = BnR-'*C. Hence, ( A u 8) n R-'*C = 0. Using a g a i n (i), (R*(A U €4)) n C = 0. Now, i f we t a k e C = % ( ( R * A ) U ( R * B ) ) , (*) i s obtained. P R O O F OF ( v i ) . B)). Hence ( v i ) . P R O O F O F ( v i i ) . By ( i i i ) , R*(A%R-'*B) 5 R*A. Also, s i n c e R - l * B n = 0 we g e t from ( i ) , ( R * ( A % R - l * B ) ) n B = 0.

We a l s o assume i n t h i s Chapter t h a t R, S, T a r e r e l a t i o n s . 1 (i) R0R-l cID(ii) R O R - ~ 510- R O R - ~ (v) R OR-^ LIDc -D I (vi) RoR-'cID- ( ~ v Ro) n R = 0. WSWT(SnT)oR = (SoR) n ( T O R ) . ( i i i ) R o R - lcID(iv) ( R o R - 1) n D v = 0 . - W S W T ( S ~ T ) =~ R ( s ~ R 2 ), (TOR). wx WY R - ~ * ( xn Y ) = ( R - ~ * x )n ( R - ~ * Y ) . V X W Y (R* X) f- Y = R*(XnR-'*Y). I t i s c l e a r t h a t o f o u r c o n s t a n t r e l a t i o n s , I D and 0 a r e f u n c t i o n s , w h i l e V x V , E L , I N , and Dv a r e n o t .