By Jörg Flum, Martin Ziegler

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E. of those e L t such t h a t (~,o1) ~ ~ and (~,o2) ~ ~ imply (~'~I n 02) ~ (resp. 6 i s the coarsest (~,01 u a 2) ~ topology c o n t a i n i n g ~1 u 02). Exercise. Show t h a t - f o r r e c u r s i v e L - the class of Lt-sentences pre- served under the i n t e r s e c t i o n (resp. union) of t o p o l o g i e s i s r e c u r s i v e l y enumerable. The r e s u l t of Lww t h a t a convex elementary class is closed under the union of s t r u c t u r e s does not g e n e r a l i z e to Lt, as i t cise.

Let X , X ' , . . set v a r i a b l e s f o r the second t o p o l o g y . Let T 1 (resp. T2) be a set of k~-sentences c o n t a i n i n g o n l y set v a r i a b l e s of the f i r s t (resp. second) s o r t such t h a t (~,~1,a2) # T i Choose new constants C o , C 1 , . . , and Vo, ~ . . . iff (~,a i ) # T . set constants U o , U 1 , . . ( f o r the second s o r t ) . ( f o r the f i r s t The set c o n s t a n t s w i l l sort), be i n t e r p r e t e d i n models by open sets o f the corresponding t o p o l o g y .

Where a l l We r e s t r i c t to many-sorted closed structures, sorts are modeh of ~, and look at logics for t h i s class of s t r u c - tures. Once more denote by ~ t t h e logic induced by the language L t . 1, i . e . 1, for • = {disc} this is e s s e n t i a l l y kindstrSmstheorem for kww. For • = @ we obtain that ~t is a "maximal" logic f o r closed s t r u c t u r e s and f o r • = {haus} we get that ~ t is also maximal i f we r e s t r i c t to t o p o l o g i c a l structures carrying Hausdorff topologies.