# Disjointed Regularity in Open Classes of Elementary Topology

Let x, y, … denote first-order structures in Stπ, x β y will denote isomorphism.

x βΌn,π y means that there is a sequence 0 =ΜΈ I0 β …. β In of sets of π-partial isomorphism of finite domain so that, for i < j β€ n, f β Ii and a β x (respectively, b β y), there is g β Ij such that g β f and a β Dom(g) (respectively, b β Im(g)). The later is called the extension property.

x βΌπ y means the above holds for an infinite chain 0 =ΜΈ I0 β …. β In β …

FraiΜsseΜβs characterization of elementary equivalence says that for finite relational vocabularies: x β‘ y iff x βΌn,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 βΌπ y implies x β y.

Given a vocabulary π let πβ be a disjoint renaming of π. If x, y β Stπ have the same power, let yβ be an isomorphic copy of y sharing the universe with x and renamed to be of type πβ. In this context, (x, yβ) will denote the π βͺ πβ-structure that results of expanding x with the relations of yβ.

Lemma: There is a vocabulary π+ β π βͺ 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 Stπ+ such that if π = ππ+,πβͺπβ then (1) π(π₯π) = {(x,yβ) : |x| = |y| β₯ π, x β‘n,π0 y}, (2) π(βn π₯n) = {(x, yβ) : |x| = |y| β₯ π, x βΌπ0 y}. Moreover, βnπ₯n is the reduct of an elementary class.

Proof. Let π₯ be the class of structures (x, yβ, <, a, I) where < is a discrete linear order with minimum but no maximum and I codes for each c β€ a a family Ic = {I(c, i, β, β)}iβx of partial π0-π0ββisomorphisms from x into yβ, such that for c < c’ β€ a : Ic β Ic and the extension property holds. Describe this by a first-order sentence ππ₯ of type π+ β π0 βͺ π0β and set π₯π = ModL(ππ₯ β§ ββ₯n x(x β€ a)}. Then condition (1) in the Lemma is granted by FraiΜsseΜβs characterization and the fact that x being (2) is granted because (x, yβ, <, a, I) β βnπ₯n iff < contains an infinite increasing π-chain below a, a β11 condition.

A topology on Stπ 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 Stπ such that the countable structures are dense in Stπ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 C1, C2 of Ξπ may be separated by an elementary class. Assume this is not the case since Ci are compact in the topology Ξπ then they are compact for the elementary topology and, by regularity of the latter, β xiΒ β CiΒ such that x1 β‘ x2 in Lππ(π). The xi must be infinite, otherwise they would be isomorphic contradicting the disjointedness of the Ci. By normality of Ξπ, there are towers Ui β Ci β Ui β Ci, i = 1,2, separating the Ci with Ui, Ui open and Ci, Ci 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 βΒ Ο and the finiteΒ π0 Β βΒ π, let t be a first-order sentence describing the commonΒ β‘n,π0 – equivalence class of x1, x2. As,

(xi,..)Β β Modπ(I)Β β©Β Ο-1 Mod(t)Β β©Β Ο-1Ui, i = 1, 2,..

and this class is open in Ξπ‘ by continuity of Ο, then by the density hypothesis there are countable xi β Ui , i = 1, 2, such that x1 β‘n,π x2. Thus for some expansion of (x1, x2β),

where π₯π,π0 is the class of Lemma, π1, π2 are reducts, and π is a renaming:

π1(x1, x2β, …) = x1 π1 : Stπ+ β Stπβͺπβ β Stπ

π2(x1, x2β, …) = x2β π2 : Stπ+ β Stπβͺπβ β Stπβ

π(x2β) = x2 π : Stπβ β Stπ

Since the classes (1) are closed by continuity of the above functors then βnπ₯n,π0 β©Β π1β1(C1) β© (ππ2)β1(C2) is non-emtpy by compactness of Ξπ+. But βnπ₯n,π0 = π(V) with V elementary of typeΒ π++Β βΒ π+. Then