Topologizable structures and Zariski topology

In this paper we study topologizability of structures. We extend the method of Kotov of topologizability of countable algebras to uncountable structures. We also show that in the case of topologizable relational countable structures the topology can be made metrizable.


Introduction
Let L = { F, R } be a countable language, where F is a family of functional symbols and R is a family of relational symbols. We denote the arity of a function f ∈ F by n f and the arity of a relation R ∈ R by n R respectively. We admit that n f may be zero; this is the case of a constant function. Let A = A, L be an L-structure. We will always assume that relations on A are not empty and have non-empty complements. Atomic formulas over A are of the form Kotov has proved in [4], that a countable algebra A is topologizable if and only if the Zariski topology of A is not discrete. This is a confirmation of the claim of A.D. Taimanov from [11]. An easy modification of Kotov's proof gives the same statement in the case of arbitrary countable language (i.e. together with relations), see Theorem 3.2 below. We remind the reader that a structure A is topologizable if it admits a non-discrete Hausdorff topology such that all operations of A are continuous and all relations are closed in the corresponding powers of A.
The problem of topologizability of algebras was initiated by A. Markov who proved in [5] a topologizability criterion for countable groups. In the sixties and seventies the problem was considered for rings (Arnautov [1]), groupoids , semigroups and skew fields (Handson and Taimanov). At the moment this topic has become well established, having well-known achievements (for example [6,8]). Paper [2,4] have nice descriptions of the topic and rich bibliography.
Let (G, ·) be a group. For every finite system of equations W (x 1 , . . . , x n ) without parameters we introduce the corresponding relational symbol R W and interpret it in G as the set of all solutions of W (x). We consider only equations having solutions and non-solutions. The structure has the same Zariski topology with (G, ·). Thus if the group G is countable it is topologizable if and only if so is the structure G REL . In particular if G is a non-topologizable group (for example the group found in [6]), then G REL is non-topolgizable too. Does this argument work in the uncountable case? This motivates us to generalize the topologizability criterion for uncountable structures with countable language. Note that in the case G REL the language is still countable even if we assume that G is uncountable.
It looks likely that structures G REL can have Hausdorff non-trivial topologies which do not topologize the group G (even when it is countable and topologized). Can these topologies be chosen with some additional properties, for example metrizability? Are there generalizations of this approach to the uncountable case? These questions are central in our paper.
In Section 3 we prove Theorem 3.2 which generalizes (and corrects the proof of) the main result of [4]. Our generalization concerns the cardinality of the structure and the presence of relations in the language. In Section 4 we prove Proposition 4.1 which shows that in the case of topologized relational countable structures the topology can be made totally disconnected.

Zariski topology
The following lemma is a folklore fact with an obvious proof. It generalizes Lemma 3 of [4] to the case structures with relations in the language. It is worth noting that following the general theory of algebraic geometry in universal algebra, Kotov defines the Zariski topology by the subbase of all sets of the form i∈I ¬ϕ i (A,ā), where I can be infinite. This is an equivalent definition.
Let τ be a topology on a set X. We remind the reader that the pseudocharacter of τ at p ∈ X is defined as follows: We will use some extension of this definition, see [2].
The following lemma is a straightforward generalization of some standard arguments, see [2,7,9]. In the following lemma we collect some easy folklore facts. Lemma 2.6. Let A be an L-structure.
(1) If τ is a Hausdorff topology on A, then any set of the form ϕ(A,ā), where ϕ is atomic, is closed in τ .

Topologizability
To prove that a structure A is topologizable we will use some strategy of defining a base of the corresponding topology. The following lemma describes this strategy. It is a generalization of Lemma 11 of [4].  be a sequence of all tuples of the form We assume that every tuple occurs cofinaly in this sequence. We enumeratē If the language L is relational, then let d be a sequence of all tuples of the form (a γ, 3 , . . . , a γ,nR+1 ). In this case we also assume that every tuple occurs cofinaly in d.
Suppose that { V γ,i } γ<κ,1≤i≤n fγ +nR γ is a family of subsets of A satisfying the following properties: Then A is topologizable and the pseudocharacter of this topology at any d ∈ D equals cf(κ).
Every element d ∈ D occurs cofinaly as a γ,1 . By (2) there is a cofinal subsequence { V γ δ ,1 } δ<cf(κ) of the corresponding γ's. As we already know it is a descending sequence of neighbourhoods of d.
The following theorem is generalization of Lemma 12 from [4] to the case of structures having relations in the language. Contrary to the Kotov's paper we do not assume that our structures are countable. This makes the construction slightly more complicated.
Then A is topologizable so that the pseudocharacter of this topology at each d ∈ D equals cf(κ).
Proof. Let d be a sequence defined in Lemma 3.1 when |A| = κ.
Let a δ,1 = d ∈ D. To satisfy that |U δ,1,δ | > 1 we construct a system of negations of atomic formulas depending on a single variable x and realized by d. When we define it we put U δ,1,δ = { d, b }, where b = d is the minimal solution of this system in the ordering b 0 , b 1 , . . . , b γ , . . . which is outside all U α,i,β with max(α, β) < δ.
We start with a family of sets of terms (depending on x) The remaining sets are constructed as follows. For every finite sequence Let us define U γ,i,δ (x) to be the union of all Uᾱ ,i,j (x) for all possible finite sequencesᾱ = { γ = α 1 < · · · < α k = δ }. Assume that b ∈ A\{ d } satisfies the condition that for any finite sequenceᾱ as above and any non-constant term t(x) from any m<j≤k a α j ,l ∈ δ <δ U αm,i,δ δ and t(d) = a γ,i then t(b) = a γ,i . Then for all γ ≤ δ and such a b, To see this note that when t(d) equals to some a γ,i as above, any its subterm of the form f δ (ā δ ) belongs to some δ <δ U β,j,δ with β < δ and a γ,i is the value of some f α , α < δ, with a substituted tuple from j δ <δ β<δ U β,j,δ .
Conditions (4),(6) and (7) are met for the family U γ,1,δ (b) if we take as b any solution of S which is outside all U α,i,β with max(α, β) < δ. Since for any α the inductive assumptions imply that d is one of the solution of this system. Since |S| < cf(κ) by Lemma 2.3(2) this system has a solution b σ = d with σ < κ such that b σ ∈ { U α,i,β | max(α, β) < δ }. The rest of the argument is clear. The assumption of the theorem, that the pseudocharacter of the Zariski topology is cf(κ) at any non-isolated point is essential. Let us assume CH and consider the non-topologizable group M of cardinality 2 ω constructed by Shelah [8]. Proof. The pseudocharacter of 1 in the Zariski topology of M is infinite. This follows from non-topologizability of M and a general observation which appears in Lemma 1 of [10]. Note that for every inequation W (x) = 1 with solution 1 any element a ∈ M is a solution of the inequation W (x · a −1 ) = 1. Thus the pseudocharacter of any element a ∈ M in the Zariski topology of M is infinite and is the same with 1. If it is 2 ω , then M is ungebunden in the sense of Podewski and thus topologizable, see [7]. As a result we see that the peudocharacter is equal to ω. Remark 3.5. Let J be the uncountable hereditarily non-topologizable group found in [3]. The term hereditarily non-topologizable means that for any H < J any quotient of H is non-topologizable. It is shown in Theorem 2.5 of [3] that there is a natural number n 0 and an element g ∈ J such that the set of nonsolutions of the equation (gg x ) n0 = 1 is finite and contains 1. Thus for any a ∈ J the equation (gg x·a −1 ) n0 = 1 is not satisfied by a. In particular any a is isolated in the Zariski topology of J. This in particular implies that the group J is not topologizable even as a relational structure J REL .

Metrizability
In the case of a countable relational structure the construction of the previous section gives a totally disconnected space. ). We may assume that any element of the tupleā m appears in this sequence as a s,1 for some s < m. This can be obtained by a small modification of the construction by duplications of some tuplesā m at some steps and introducing several steps in the beginning where we do nothing. When a s,1 ∈ D and a s,1 was not used before step s as some b in order to satisfy condition (5), then the construction of Theorem 3.2 ensures that all V n,i containing a s,1 are singletons. Moreover note that any element a s,1 ∈ D first appearing as such a b defines singletons V n,i for a n,i = a s,1 from the moment of the second appearance.
Let us consider the corresponding V n,i for elements of D. Since L is relational the procedure of Theorem 3.2 in this case can be equivalently reformulated in a more convenient way. We now describe it.
Let a s,1 = d ∈ D. To satisfy that |U s,1,s | > 1 we construct a system of negations of atomic formulas depending on a single variable x and realized by d. When we define it we put U s,1,s = { d, b }, where b = d is the minimal solution of this system in the enumeration b 0 , b 1 , . . . , b n , . . . which is outside all U k,i,l with max(k, l) < s. The assumptions of the beginning of the proof guarantee that b does not belong to the set { a s,i | i ≤ n Rs + 1 }.
It is easy to see that these rules agree with the corresponding procedure of Theorem 3.2.
Since As a result we see that in the case a s,1 ∈ D the procedure of step s only extends the sets U m,i,s−1 which contain a s,1 . Moreover these sets are extended by the same element which is outside all U m,i,k with m ≤ s and k < s.
Applying easy induction we obtain that for any (m, i, k) and (n, j, k) with m < n and k ≤ s if the intersection U m,i,k ∩ U n,j,k is not empty, then U m,i,k contains U n,j,k . This implies the following claim.
Claim. For any (m, i) and (n, j) with m < n if the intersection V m,i ∩ V n,j is not empty, then V m,i contains V n,j .
To see that the topology is zero dimensional it suffices to show that all sets of the form V n,1 are clopen. Let F be the complement of some V n,1 . Any a ∈ F coincides with some a m,1 with m > n. By our claim the corresponding V m,1 has empty intersection with V n,1 .
Thus the set F can be presented as a union of basic open sets, i.e. F is clopen.
Since the family of all V m,i is countable, by the Urysohn's metrization theorem we have that the topology τ is metrizable. The rest of the formulation follows from Theorem 3.2.
Let us mention that in the situation when G is a countable topological group we cannot state that the topology obtained for G REL by the method of Proposition 4.1 topologizes the group G. This is because the construction from the proof does not take care of continuity of multiplication.