Let x, y, … denote first-order structures in St_{π}, x β y will denote isomorphism.

x βΌ_{n,π} y means that there is a sequence 0 =ΜΈ I_{0} β …. β I_{n} of sets of π-partial isomorphism of finite domain so that, for i < j β€ n, f β I_{i} and a β x (respectively, b β y), there is g β I_{j} such that g β f and a β Dom(g) (respectively, b β Im(g)). The later is called the extension property.

x βΌ_{π} y means the above holds for an infinite chain 0 =ΜΈ I_{0} β …. β I_{n} β …

* FraiΜsseΜβs characterization of elementary equivalence* says that for finite relational vocabularies: x β‘ y iff x βΌ

_{n,π}y. To have it available for vocabularies containing function symbols add the complexity of terms in atomic formulas to the quantifier rank. It is well known that for countable x, y : x βΌ

_{π}y implies x β y.

Given a vocabulary π let π^{β} be a disjoint renaming of π. If x, y β St_{π} have the same power, let y^{β} be an isomorphic copy of y sharing the universe with x and renamed to be of type π^{β}. In this context, (x, y^{β}) will denote the π βͺ π^{β}-structure that results of expanding x with the relations of y^{β}.

Lemma: There is a vocabulary π^{+} β π βͺ π^{β} such that for each finite vocabulary π_{0} β π there is a sequence of elementary classes π₯_{1} β π₯_{2} β π₯_{3} β …. in St_{π+} such that if π = π_{π+,πβͺπβ} then (1) π(π₯_{π}) = {(x,y^{β}) : |x| = |y| β₯ π, x β‘_{n,π0} y}, (2) π(β_{n} π₯_{n}) = {(x, y^{β}) : |x| = |y| β₯ π, x βΌ_{π0} y}. Moreover, β_{n}π₯_{n} is the reduct of an elementary class.

Proof. Let π₯ be the class of structures (x, y^{β}, <, a, I) where < is a discrete linear order with minimum but no maximum and I codes for each c β€ a a family I_{c} = {I(c, i, β, β)}_{iβx} of partial π_{0}-π_{0}^{β}βisomorphisms from x into y^{β}, such that for c < c’ β€ a : I_{c} β I_{c} and the extension property holds. Describe this by a first-order sentence π_{π₯} of type π^{+} β π_{0} βͺ π_{0}^{β} and set π₯_{π} = Mod_{L}(π_{π₯} β§ β^{β₯n} x(x β€ a)}. Then condition (1) in the Lemma is granted by FraiΜsseΜβs characterization and the fact that x being (2) is granted because (x, y^{β}, <, a, I) β β_{n}π₯_{n} iff < contains an infinite increasing π-chain below a, a β_{1}^{1} condition.

A topology on St_{π} is invariant if its open (closed) classes are closed under isomorphic structures. Of course, it is superfluous if we identify isomorphic structures.

Theorem: Let Ξ be a regular compact topology finer than the elementary topology on each class St_{π} such that the countable structures are dense in St_{π}Β and reducts and renamings are continuous for these topologies. Then Ξ_{π} is the elementary topology β π.

Proof: We show that any pair of disjoint closed classes C_{1}, C_{2} of Ξ_{π} may be separated by an elementary class. Assume this is not the case since C_{i} are compact in the topology Ξ_{π} then they are compact for the elementary topology and, by regularity of the latter, β x_{i}Β β C_{i}Β such that x_{1} β‘ x_{2} in L_{ππ}(π). The x_{i} must be infinite, otherwise they would be isomorphic contradicting the disjointedness of the C_{i}. By normality of Ξ_{π}, there are towers U_{i} β C^{‘}_{i} β U^{‘}_{i} β C^{”}_{i}, i = 1,2, separating the C_{i} with U_{i}, U^{‘}_{i} open and C^{‘}_{i}, C^{”}_{i} closed in Ξ_{π} and disjoint. Let I be a first-order sentence of type π^{‘} β π such that (z, ..) |= I β z is infinite, and letΒ Ο be the corresponding reduct operation. For fixed nΒ βΒ Ο and the finiteΒ π_{0} Β βΒ π, let t be a first-order sentence describing the commonΒ β‘_{n,π0} – equivalence class of x_{1}, x_{2}. As,

(x_{i},..)Β β Modπ^{‘}(I)Β β©Β Ο^{-1} Mod(t)Β β©Β Ο^{-1}U_{i}, i = 1, 2,..

and this class is open in Ξ_{π}‘ by continuity of Ο, then by the density hypothesis there are countable x_{i} β U_{i} , i = 1, 2, such that x_{1} β‘_{n,π} x_{2}. Thus for some expansion of (x_{1}, x_{2}^{β}),

(x, x^{β},..) β π₯_{n,π0} β© π_{1}^{β1}(πΆ^{‘}_{1}) β© (ππ_{2})^{β1}(C^{‘}_{2}) —– (1)

where π₯_{π,π0} is the class of Lemma, π_{1}, π_{2} are reducts, and π is a renaming:

π_{1}(x_{1}, x_{2}^{β}, …) = x_{1} π_{1} : St_{π+} β St_{πβͺπβ} β St_{π}

π_{2}(x_{1}, x_{2}^{β}, …) = x_{2}^{β} π_{2} : St_{π+} β St_{πβͺπβ} β St_{πβ}

π(x_{2}^{β}) = x_{2} π : St_{πβ} β St_{π}

Since the classes (1) are closed by continuity of the above functors then β_{n}π₯_{n,π0} β©Β π_{1}^{β1}(C^{‘}_{1}) β© (ππ_{2})^{β1}(C^{‘}_{2}) is non-emtpy by compactness of Ξ_{π+}. But β_{n}π₯_{n,π0} = π(V) with V elementary of typeΒ π^{++}Β βΒ π^{+}. Then

VΒ β©Β Ο^{-1}Ο_{1}^{-1}(U^{‘}_{1})Β β©Β Ο^{-1}(ΟΟ_{2})^{-1} (U^{‘}_{2})Β β 0

is open forΒ Ξ_{L++} and the density condition it must contain a countable structure (x_{1}, x^{*}_{2}, ..). ThusΒ (x_{1}, x^{*}_{2}, ..)Β βΒ β©_{n}Β π₯_{π,π0}, with x_{i}Β β U^{‘}_{i}Β β C^{”}_{i}. It follows that x_{1} ~_{π0} x_{2} and thus x_{1} |π_{0}Β β x_{2} |π_{0}. LetΒ Ξ΄_{π0} be a first-order sentence of typeΒ πΒ βͺΒ π^{*}Β βͺ{h} such that (x, y^{*}, h) |= Ξ΄_{π0}Β β h : x |π_{0}Β β y|π_{0}. By compactness,

(β©_{π0}β_{fin}π Mod_{πβͺπ*βͺ{f}} (Ξ΄_{π0}))Β β©Β Ο_{1}^{-1}(C^{”}_{1})Β β© (ΟΟ_{2})^{-1}(C^{”}_{2})Β β 0

and we have h : x_{1}Β β x_{2}, x_{i}Β β C^{”}_{i}, contradicting the disjointedness of C^{”}_{i}. Finally, if C is a closed class ofΒ Ξ_{π} and xΒ β C, cl_{Ξπ{x}} is disjoint from C by regularity ofΒ Ξ_{π}. Then cl_{Ξπ{x}} and C may be separated by open classes of elementary topology, which implies C is closed in this topology.