Disjointed Regularity in Open Classes of Elementary Topology

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Let x, y, … denote first-order structures in St𝜏, x β‰ˆ y will denote isomorphism.

x ∼n,𝜏 y means that there is a sequence 0 =ΜΈ I0 βŠ† …. βŠ† In of sets of 𝜏-partial isomorphism of finite domain so that, for i < j ≀ n, f ∈ Ii and a ∈ x (respectively, b ∈ y), there is g ∈ Ij such that g βŠ‡ f and a ∈ Dom(g) (respectively, b ∈ Im(g)). The later is called the extension property.

x ∼𝜏 y means the above holds for an infinite chain 0 =ΜΈ I0 βŠ† …. βŠ† In βŠ† …

Fraïssé’s characterization of elementary equivalence says that for finite relational vocabularies: x ≑ y iff x ∼n,𝜏 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 ∼𝜏 y implies x β‰ˆ y.

Given a vocabulary 𝜏 let πœβˆ— be a disjoint renaming of 𝜏. If x, y ∈ St𝜏 have the same power, let yβˆ— be an isomorphic copy of y sharing the universe with x and renamed to be of type πœβˆ—. In this context, (x, yβˆ—) will denote the 𝜏 βˆͺ πœβˆ—-structure that results of expanding x with the relations of yβˆ—.

Lemma: There is a vocabulary 𝜏+ βŠ‡ 𝜏 βˆͺ πœβˆ— such that for each finite vocabulary 𝜏0 βŠ† 𝜏 there is a sequence of elementary classes π›₯1 βŠ‡ π›₯2 βŠ‡ π›₯3 βŠ‡ …. in St𝜏+ such that if πœ‹ = πœ‹πœ+,𝜏βˆͺπœβˆ— then (1) πœ‹(π›₯𝑛) = {(x,yβˆ—) : |x| = |y| β‰₯ πœ”, x ≑n,𝜏0 y}, (2) πœ‹(β‹‚n π›₯n) = {(x, yβˆ—) : |x| = |y| β‰₯ πœ”, x ∼𝜏0 y}. Moreover, β‹‚nπ›₯n is the reduct of an elementary class.

Proof. Let π›₯ be the class of structures (x, yβˆ—, <, a, I) where < is a discrete linear order with minimum but no maximum and I codes for each c ≀ a a family Ic = {I(c, i, βˆ’, βˆ’)}i∈x of partial 𝜏0-𝜏0βˆ—β€“isomorphisms from x into yβˆ—, such that for c < c’ ≀ a : Ic βŠ† Ic and the extension property holds. Describe this by a first-order sentence πœƒπ›₯ of type 𝜏+ βŠ‡ 𝜏0 βˆͺ 𝜏0βˆ— and set π›₯𝑛 = ModL(πœƒπ›₯ ∧ βˆƒβ‰₯n x(x ≀ a)}. Then condition (1) in the Lemma is granted by Fraïssé’s characterization and the fact that x being (2) is granted because (x, yβˆ—, <, a, I) ∈ β‹‚nπ›₯n iff < contains an infinite increasing πœ”-chain below a, a βˆ‘11 condition.

A topology on St𝜏 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 St𝜏 such that the countable structures are dense in St𝜏 and reducts and renamings are continuous for these topologies. Then Ξ“πœ is the elementary topology βˆ€ 𝜏.

Proof: We show that any pair of disjoint closed classes C1, C2 of Ξ“πœ may be separated by an elementary class. Assume this is not the case since Ci are compact in the topology Ξ“πœ then they are compact for the elementary topology and, by regularity of the latter, βˆƒ xi ∈ CiΒ such that x1 ≑ x2 in Lπœ”πœ”(𝜏). The xi must be infinite, otherwise they would be isomorphic contradicting the disjointedness of the Ci. By normality of Ξ“πœ, there are towers Ui βŠ† Ci βŠ† Ui βŠ† Ci, i = 1,2, separating the Ci with Ui, Ui open and Ci, Ci closed in Ξ“πœ and disjoint. Let I be a first-order sentence of type 𝜏 βŠ‡ 𝜏 such that (z, ..) |= I ⇔ z is infinite, and letΒ Ο€ be the corresponding reduct operation. For fixed nΒ βˆˆΒ Ο‰ and the finite 𝜏0 Β βŠ†Β πœ, let t be a first-order sentence describing the common ≑n,𝜏0 – equivalence class of x1, x2. As,

(xi,..) ∈ Mod𝜏(I)Β βˆ©Β Ο€-1 Mod(t)Β βˆ©Β Ο€-1Ui, i = 1, 2,..

and this class is open in Ξ“πœ‘ by continuity of Ο€, then by the density hypothesis there are countable xi ∈ Ui , i = 1, 2, such that x1 ≑n,𝜏 x2. Thus for some expansion of (x1, x2βˆ—),

(x, xβˆ—,..) ∈ π›₯n,𝜏0 ∩ πœ‹1βˆ’1(𝐢1) ∩ (πœŒπœ‹2)βˆ’1(C2) —– (1)

where π›₯𝑛,𝜏0 is the class of Lemma, πœ‹1, πœ‹2 are reducts, and 𝜌 is a renaming:

πœ‹1(x1, x2βˆ—, …) = x1 πœ‹1 : St𝜏+ β†’ St𝜏βˆͺπœβˆ— β†’ St𝜏

πœ‹2(x1, x2βˆ—, …) = x2βˆ— πœ‹2 : St𝜏+ β†’ St𝜏βˆͺπœβˆ— β†’ Stπœβˆ—

𝜌(x2βˆ—) = x2 𝜌 : Stπœβˆ— β†’ St𝜏

Since the classes (1) are closed by continuity of the above functors then β‹‚nπ›₯n,𝜏0 βˆ©Β πœ‹1βˆ’1(C1) ∩ (πœŒπœ‹2)βˆ’1(C2) is non-emtpy by compactness of Ξ“πœ+. But β‹‚nπ›₯n,𝜏0 = πœ‹(V) with V elementary of type 𝜏++Β βŠ‡Β πœ+. Then

VΒ βˆ©Β Ο€-1Ο€1-1(U1)Β βˆ©Β Ο€-1(ρπ2)-1 (U2)Β β‰  0

is open forΒ Ξ“L++ and the density condition it must contain a countable structure (x1, x*2, ..). ThusΒ (x1, x*2, ..) ∈ ∩nΒ π›₯𝑛,𝜏0, with xi ∈ UiΒ βŠ† Ci. It follows that x1 ~𝜏0 x2 and thus x1 |𝜏0Β β‰ˆ x2 |𝜏0. Let δ𝜏0 be a first-order sentence of type 𝜏 βˆͺ 𝜏*Β βˆͺ{h} such that (x, y*, h) |= δ𝜏0 ⇔ h : x |𝜏0Β β‰ˆ y|𝜏0. By compactness,

(∩𝜏0βŠ†fin𝜏 Mod𝜏βˆͺ𝜏*βˆͺ{f} (δ𝜏0))Β βˆ©Β Ο€1-1(C1) ∩ (ρπ2)-1(C2)Β β‰  0

and we have h : x1Β β‰ˆ x2, xi ∈ Ci, contradicting the disjointedness of Ci. Finally, if C is a closed class ofΒ Ξ“πœ and xΒ βˆ‰ C, clΞ“πœ{x} is disjoint from C by regularity ofΒ Ξ“πœ. Then clΞ“πœ{x} and C may be separated by open classes of elementary topology, which implies C is closed in this topology.

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