By Jörg Flum, Martin Ziegler

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For L : {Co,Cl} let ~o : 3×~ co 3Y ~ c I Vz(~ z e X v ~ z ~Y). Show: (a) Given ($,~) ~ ~o and substructures (~i, ai),(~2,~2) and (~3,o3) of (~,a)~ i f (~1,al) # ~o' (~2'02) ~ @o and A1 n A2 = A3, then (~3,o3) ~ @o" (b) There i s o sequence (~n,On) of models of ~o with (~n'°n) c (~n~q'On+l) such t h a t the union of the (~n, an) ~ . e . of t o p o l o g i c a l the injective limit in the category spaces) i s not a model of ~ . 0 § 7 Definability First we show t h a t some theorems on the explicit definability of relations generalize from Lww to L t.

M1,N 1 . . . Nm are d e f i n e d ) k 3Y 9 t ~ ' [ ~ , O , V ] . i(~i,oi "matrix" (4) ~ X[M1 . . . s = {jli by (1) and ( 2 ) . We show t h a t By (3) we f i n d Z h, 1 -< h _< 1, s a t i s f y i n g ~ j=l e Zj, = )C[Z] ..... (Zj - N j ) , Zm]. l e t V i = A 1. For i e 1 -< j < m}. Since Z h c Mh t h e r e V. • a. w i t h 1 the of X. In p a r t i c u l a r ('~(I),Fin) For i ~ Nm] 1 t(~i'~i)[~(i)] ~ vi a~d (~i,~i) ~ ^ ,j[~(il,O(il,~(il,Vi]. Jes h is 36 Let V be g V.. Then t i~I I ~(~/i,°i ) [a] e V m L3 (Zj - Nj) and since is finite, j =I o Vega..

Then I1 (~I'~I) /D I Z. (~i'~i) t ]/D. ~ (~2,~2) 12 /D2 and, by ~os theorem, (~i'~i) for i = 1,2. Hence (~1,~1)~ t (~2,~2). - ] Now assume that (~1,~1) ~t(~2,~2). 2 and ( ~ i , ~ i ) ~ (~,T i) for i = 1,2. By the Keisler- Shelah-theorem we find an u l t r a f i l t e r (~i,a])I/D ~ (~,T])I/D D over a set I such that for i = 1,2. Put (~*,~i*) = (~,T])I/D . Since (~,Tl) : (~,~2), we have (~*,TT) : (~*,~). Hence (~l,al)I/D ~ ( ~ * , T ~ ) t (~*,T~)~ (~2,a2)I/D. 19 Theorem. Each L2-sentence invoriant for topologies ks equivalent in topological structures to an Lt-sentence, i .