Theory and Applications of Categories, Vol. 25, No. 19, 2011, pp. 508–532.

COUNTABLE MEETS IN COHERENT SPACES WITH APPLICATIONS TO THE CYCLIC SPECTRUM

MICHAEL BARR, JOHN F. KENNISON, AND R. RAPHAEL

Abstract. This paper reviews the basic properties of coherent spaces, characterizes them, and proves a theorem about countable meets of open sets. A number of examples of coherent spaces are given, including the set of all congruences (equipped with the Zariski topology) of a model of a theory based on a set of partial operations. We also give two alternate proofs of the main theorem, one using a theorem of Isbell’s and a second using an unpublished theorem of Makkai’s. Finally, we apply these results to the Boolean cyclic spectrum and give some relevant examples.

1. Introduction

A frame is a complete lattice in which finite infs distribute over arbitrary sups. We denote the empty inf by ⊤ and the empty sup by ⊥, which are the top and bottom elements, respectively, of the lattice. A map of frames preserves finite infs and arbitrary sups. The motivating example of a frame is the open set lattice of a topological space. Moreover, continuous maps induce frame homomorphisms. The result is a contravariant functor O from the category Topof topological spaces to the category Frmof frames. A closed subset D of a topological space is called indecomposable if it is not possible to write it as a union of two proper closed subsets. A space is called sober if every indecomposable closed set is the closure of a unique point, called the generic point of the set. On sober spaces O is full and faithful.

If we let Locdenote the category of locales, which is simply Frm^{op}, the opposite of the category of frames, this results in a covariant functor Top [figure] Loc.

In Section 2, we review basic properties of coherent spaces and prove a characterization theorem which is similar to known results. Section 3 shows several ways in which coherent spaces arise. A notable example concerns models of a first order theory described by operations and partial operations. We show, for example, that the set of subobjects as well as the set of congruences of a model, when equipped with a certain topology, called the Zariski topology, give coherent spaces. In Section 4 we state and prove the main theorem that shows that if X is a coherent space and {U_{i}} a countable family of open

The first author would like to thank NSERC of Canada for its support of this research. We would all like to thank McGill and Concordia Universities for partial support of the second author’s visits to Montreal.

Received by the editors 2011-06-14 and, in revised form, 2011-10-16.

Transmitted by Susan Niefeld. Published on 2011-11-19.

2000 Mathematics Subject Classification: 06D22, 18B99, 37B99.

Key words and phrases: countable localic meets of subspaces, Boolean flows, cyclic spectrum.

\begin{matrix}c \\ ◯ \end{matrix} Michael Barr, John F. Kennison, and R. Raphael, 2011. Permission to copy for private use granted.

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subsets of X then the intersection ⋂ U_{i} in the lattice of subspaces of X coincides with the inf ⋀ U_{i} in the lattice of sublocales of X. Section 5 discusses the connection with an unpublished theorem of Makkai’s. In Section 6, we apply our results to the Boolean cyclic spectrum and thus extend the work in [Kennison, 2002, Kennison, 2006, Kennison, 2009]. Section 7 gives examples.

1.1. Remark. In dealing with locales, it is standard to use “sublocale” to mean regular subobject. This means that sublocales correspond to regular quotients of frames. Since Frmis equational, there is a one-one correspondence between regular quotients and equivalence relations that are also models of the theory. Such equivalence relations are called congruences. Thus if F is a frame there is a one-one correspondence between congruences on F and sublocales of the locale L corresponding to F.

When we spoke on Theorem 4.1 of this paper at Category Theory 2011 in Vancouver, André Joyal conjectured that this result was closely related to the property of being a Baire space. We already knew that a space that satisfied Theorem 4.1 was Baire and had an example of a Baire space that didn’t satisfy the theorem. So it was no surprise that in the process of searching the literature, we discovered a paper by Till Plewe that showed that the localic inf of a sequence of open sets in a space is spatial if and only if every closed subset of X is Baire [Plewe (1996), Theorem 2.3]. In addition, we discovered a theorem of John Isbell’s that shows that a locally compact (he called them locally quasicompact) locale and intersections of descending sequences of such sublocales are spatial [Isbell (1975), 4.1]. This result comes very close to proving our Theorem 4.1 as we will see in the discussion following our proof. We feel that Theorem 4.1 and its related results deserve to be better known because of their potential application to spectra, such as the cyclic spectrum of a Boolean flow. We also mention some extensions of the theorem, answering another question raised by Joyal, see 4.8.

2. Basic definitions and preliminary results

The results in this section are all standard and can be found, with somewhat different proofs, in [Johnstone, 1982].

2.1. congruences and nuclei. A nucleus j on a frame F is a function j : F [figure] F such that

Nuc-1. j is expansive: u ≤ j(u) for all u ∈ F;

Nuc-2. j preserves finite inf;

Nuc-3. j is idempotent.

2.2. Theorem. There is a one-one correspondence between nuclei and congruences on a frame.

Proof. Let F be a frame and j be a nucleus on F. Define a relation E by u E v if j(u) = j(v). Since this clearly defines an equivalence relation, it is sufficient to show it is closed under the frame operations. If u_{1} E v_{1} and u_{2} E v_{2}, it follows immediately from Nuc-2, that (u_{1} ∧ u_{2}) E (v_{1} ∧ v_{2}). One might expect that showing that E is closed under arbitrary sup would require that j preserve arbitrary sup, which it does not do in general. From Nuc-2, it is clear that j is order preserving. Suppose we have two families {u_{α}} and {v_{α}} such that u_{α} E v_{α} for all α. Then u_{α} ≤ j(u_{α}) = j(v_{α}) ≤ j(⋁ v_{α}). Thus ⋁ u_{α} ≤ j(⋁ v_{α}) so that j(⋁ u_{α}) ≤ j^{2}(⋁ v_{α}) = j(⋁ v_{α}) and the opposite inequality follows by symmetry. Thus E is a congruence.

Now suppose that E is a congruence on F. Define j by j(u) = ⋁{v | u E v}. It is clear that j is expansive. Since E is closed under arbitrary sup, it is also clear that u E j(u) from which it follows that j(u) E j^{2}(u) so that j^{2}(u) ≤ j(u). To see that j preserves finite inf, we first show it preserves the partial order. If u_{1} ≤ u_{2}, then for all v with v E u_{1}, we have (v ∨ u_{2}) E (u_{1} ∨ u_{2}) = u_{2} so that v ≤ v ∨ u_{2} ≤ j(u_{2}). Since this is true of all such v, it is true of their union which is j(u_{1}). It is immediate that j(u_{1} ∧u_{2}) ≤ j(u_{1})∧j(u_{2}). Next, (u_{1} ∧u_{2}) E (j(u_{1})∧j(u_{2})) and, by definition of j, it follows that j(u_{1} ∧u_{2}) ≥ j(u_{1})∧j(u_{2}).

It is easy to verify that these processes are inverse to each other.

2.3. Coherent spaces. A topological space is said to be coherent if it is compact, sober, the compact open sets are a base for the topology, and the intersection of two compact open sets is compact.

If X is a topological space and M is a subbase for the topology on X, let N be the set of complements of sets in M . We call the topology generated by M ∪ N the s-topology (for strong topology). We will say that a subset of X is s-open, s-closed, or scompact, respectively, if it is open, closed, or compact, respectively, in the s-topology. It is clear that open and closed sets are s-open and s-closed, respectively, while an s-compact set is compact.

This s-topology as defined is an example of a topology gotten by beginning with a subbase and adjoining the complements of the elements to get a larger subbase. Such a topology is called a patch topology and we will adopt this terminology. We will sometimes call the original topology the w-topology (for weak). We note that this construction of a patch topology depends on the given subbase and is thus not a topological construction. For example, on a Stone space, you recover the original topology if you take for a subbase all the clopens, but if you take all opens, you just get the discrete topology.

2.4. Theorem. A topological space X is coherent if and only if it has a subbase M consisting of compact sets and such that the topology generated by M and the complements of the sets in M is compact Hausdorff.

Proof. The forward implication is based on the proof of [Hochster, 1969, Theorem 1]. Let M denote the set of all compact open sets. The definition of coherent implies that M is closed under finite meets and it is obviously closed under finite joins. This implies that N is closed under finite meet and join as well. Since M ∪N is closed under complementation it is also a subbase for the closed set lattice in the s-topology, By dualizing [Kelley, 1955,

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Theorem 4.6], it will suffice to show that for any M_{0} ⊆ M and any N_{0} ⊆ N , if M_{0} ∪ N_{0} has the finite intersection property (FIP), then it has a non-empty meet. We will do this using a series of claims.

We can assume that M_{0} is closed under finite meets and that N_{0} is maximal. The first is trivial, while the second follows readily from the fact that the join of any chain of families with the FIP has that property, since the FIP is determined by the finite subfamilies.

The meet D = ⋂_{N∈N_{0}} N ≠ ∅ and meets every M ∈ M_{0} so that the family M_{0} ∪ {D} has the FIP. Fix M ∈ M_{0}. The family {M ∩ N | N ∈ N_{0}} certainly has the FIP and is a family of s-closed subsets of the compact space M.

D is indecomposable. Suppose D = D_{1} ∪ D_{2} with D_{1} and D_{2} closed subsets of D. At least one of M_{0} ∪ {D_{1}} and M_{0} ∪ {D_{2}} has the FIP. Suppose that M_{0} ∪ {D_{1}} has the FIP. Since D_{1} is closed, it is an intersection of sets in N . These sets can be added to N_{0} without destroying the FIP in M_{0} ∪ N_{0} and, by maximality, must already belong to N_{0}. But this implies that D_{1} ⊇ ⋂_{N∈N_{0}} N and hence D = D_{1}.

The generic point x of D is in every M ∈ M_{0}. For if x ∉ M ∈ M_{0}, then D − M would be a proper closed subset of D that contained x.

This completes the proof of the forward implication. For the converse, we begin by noting that the sets in M are s-closed and hence s-compact and the same is true of their finite meets since, by hypothesis, the s-topology is Hausforff. So we may suppose that M is closed under finite meets and joins. By assumption, M is a base for the topology and we easily see that every compact open set belongs to M .

Since X is s-compact, it is also compact. In view of the above definition of coherent spaces, it suffices to show that X is sober. It is clear that X is T_{0} for if x, y ∈ X are such that for every M ∈ M we have x ∈ M if and only if y ∈ M, then the same is true for the family consisting of all sets in M ∪ N . Since this family forms a base for the s-topology, which is Hausdorff, it follows that x = y. We denote by x̄, the closure of {x}. Let A be a closed, indecomposable subset. We have to find a point p, necessarily in A, such that A is \overline{p}. Since X is T_{0}, such a point is unique if it exists. Assume that no such point p exists. Then for every a ∈ A, we can choose a point φ(a) ∈ A such that φ(a) ∉ \overline{a}. Since φ(a) ∉ \overline{a}, there exists a basic neighbourhood, M_{a} ∈ M , of φ(a) which misses a. Then a is in ¬M_{a}, the complement of M_{a}. Since ¬M_{a} is s-open and A is s-closed, hence s-compact, there is a finite subset F ⊆ A such that A is covered by {¬M_{a} | a ∈ F }. Assume that F is chosen as small as possible. The set F cannot consist of a single element since φ(a) ∉ ¬M_{a}. But if F = F_{1}∪F_{2} is the union of two non-empty subsets, then A ⊆ (⋃_{a∈F_{1}} ¬M_{a})∪(⋃_{a∈F_{2}} ¬M_{a}). But these are closed sets and A is indecomposable, so it must be contained in the one or the other set. This contradicts the assumption that F was chosen as small as possible.

2.5. Proposition. Suppose X is a coherent space with base M of compact open sets. Suppose {M_{α}} is a family of sets from M and U is an open subset of X. If ⋂ M_{α} ⊆ U, then for some finite set, say α_{1}, …, α_{m} of indices, we have that ⋂_{i=1}^{m} M_{α_{i}} ⊆ U.

Proof. The sets M_{α} are s-closed in a compact space. The set U is open, hence s-open and therefore each M_{α} − U is closed. If ⋂ M_{α} ⊆ U, then ⋂(M_{α} − U) = ∅, whence a finite intersection of them is empty.

3. Examples of Coherent Spaces

This section shows that coherent spaces arise in many ways. Often the proof that a given space is coherent is omitted because it easily follows from the definition or from Theorem 2.4.

3.1. Notation. Whenever X is a given coherent space, M will denote the base of all compact open subsets and N will denote the family of all sets whose complements are in M . When constructing a coherent space, M will denote a family satisfying the conditions of Theorem 2.4 and, after closing M up under finite joins and meets, N will denote the family of all sets whose complements are in M .

3.2. Example. Any s-closed subspace of a coherent space is coherent.

3.3. Example. Let X be coherent and let M and N be as above. Then X with the topology generated by N is coherent. We call the topology generated by N the dual of the original topology, generated by M .

3.4. Definition. Let S be any set and let 2^{S} be the family of all subsets of S. For each a ∈ S let M(a) = {A ⊆ S | a ∈ A} and M = {M(a) | a ∈ S}. Then:

1. the w-topology on 2^{S} is the one generated by the subbase M ;

2. the s-topology on 2^{S} is generated by M together with N , the family of all complements of members of M ;

3. if F ⊆ 2^{S} then the w-topology (respectively the s-topology) on F is the relative topology on F obtained from the w-topology (resp. the s-topology) on 2^{S}.

3.5. Example. The space 2^{S}, with the w-topology, is coherent.

In general, if F ⊆ 2^{S} is an s-closed subset, then F , with the w-topology, is also coherent.

Proof. The w-topology on 2^{S} is generated by M as defined above. In view of Theorem 2.5, it suffices to observe that the s-topology, generated by M ∪N is the product topology, obtained by regarding 2^{S} as a product of S copies of 2 (where 2 is the discrete space with two points). The assertion about F follows from 3.2.

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3.6. Notation. If U is an ultrafilter on 2^{S}, then A_{U} ⊆ S denotes the subspace for which a ∈ A_{U} if and only if M(a) ∈ U.

3.7. Example. Let F ⊆ 2^{S} be given and suppose that A_{U} ∈ F whenever U is an ultrafilter on 2^{S} with F ∈ U. Then F , with the w-topology, is coherent.

Proof. It suffices to observe that A_{U} is the limit of U in the s-topology on 2^{S}. So the given condition implies that F is an s-closed subset of 2^{S}.

3.8. Definition. Let T be a first order theory. We will say it is generated by finitary partial operations if there is a family Ω = {Ω_{0}, Ω_{1}, …, Ω_{n}, …} of sets such that an algebra S for T is given by a partial function ωS : S^{n} [figure] S for each n ∈ N and each ω ∈ Ω_{n}. These partial operations may be subject to equations and Horn clauses, but they play no role in the construction.

3.9. Example. Let S be a T-algebra. Let F be the family of all subsets of F ⊆ S which satisfy (finitary) first-order conditions built up from equality, the operations of T, the conditions x ∈ F and closed under binary infs, sups and negation. Then F , with the w-topology, is coherent.

Such families would include T-subalgebras and, T-congruences, in case S = R × R where R is a model of a T-algebra.

Proof. Such a family F satisfies the condition that whenever U is an ultrafilter on 2^{S} and F ∈ U, then A_{U} ∈ F .

3.10. Example. If R is a ring, then the set P of all prime ideals of R, with the w-toplogy, is coherent. The dual of this space is the set P with the Zariski topology.

3.11. Remark. It is shown in [Hochster, 1969] that every coherent space X is homeomorphic to a space of the form P (as in the above example, with the w-topology) and also homeomorphic to a space P with the Zariski topology.

4. Countable meets

Recall from 1.1 that a sublocale is a regular subobject in the category of locales and that there is a one-one correspondence between nuclei and congruences on a frame. If the frame is O(X), the lattice of open sets of a space X, and if A ⊆ X is a subspace, then the nucleus j_{A} defined by j_{A}(U) = ⋃{V ∈ O(X) | V ∩ A ⊆ U} corresponds to the subframe O(A).

4.1. Theorem. Suppose that X is coherent and suppose {U_{i}} is a countable sequence of open sets. Then their spatial intersection ⋂ U_{i} coincides with their localic inf ⋀ U_{i}.

Proof. Let M be the base of compact open sets. Suppose A = ⋂ U_{n} and that U_{n} = ⋃_{σ∈Σ_{n}} M_{n,σ} with each M_{n,σ} ∈ M . Let L = ⋀ U_{n} and denote by j_{n}, J_{A}, and j_{L}, resp. the nuclei corresponding to U_{n}, A, and L. By definition, j_{L} = ⋁ j_{n}, the sup taken in the lattice of nuclei. Since A ⊆ U_{n} for all n, we see that j_{n} ≤ j_{A} whence j_{L} ≤ j_{A}.

By a choice function, we mean a map ξ : N [figure]⋃ Σ_{n} such that ξ(n) ∈ Σ_{n} for all n > 0. If ξ is a choice function, we let A_{ξ} = ⋂_{n=1}^{∞} M_{n,ξ(n)}. Then from M_{n,ξ(n)} ⊆ U_{n}, it follows that A_{ξ} ⊆ A.

If we suppose that A ≨ L, (in the lattice of sublocales of X) then j_{L} ≨ j_{A}. Thus there is an open set V such that j_{L}(V) ⫋ j_{A}(V) and hence there is an M_{0} ∈ M with M_{0} ⊆ j_{A}(V) while M_{0} ⊈ j_{L}(V). This last implies that for all n > 0, M_{0} ⊈ j_{n}(V) which, we will show, leads to a contradiction.

4.2. Lemma. Suppose that M ∈ M with M ⊈ j_{L}(V). Then for each n > 0, there is a σ ∈ Σ_{n} such that M ∩ M_{n,σ} ⊈ j_{L}(V).

Since M ⊈ j_{L}(V) = j_{L}^{2}(V) and j_{L} = ⋁ j_{n}, we see that M ⊈ j_{n}(j_{L}(V)) and hence M ∩ U_{n} ⊈ j_{L}(V). But U_{n} = ⋃_{σ∈Σ_{n}} M_{n,σ} so there must be some σ ∈ Σ_{n} with M ∩ M_{n,σ} ⊈ j_{L}(V).

We will use this lemma to construct a choice function ξ such that M_{0} ∩ A_{ξ} ⊈ j_{L}(U). Assuming this can be done, it follows from A_{ξ} ⊆ A that M_{0} ∩ A = M_{0} ∩ ⋃_{ξ∈Ξ} A_{ξ} ⊈ U from which we conclude that M_{0} ⊈ j_{A}(U) in contradiction to our supposition.

In this proof we use the standard notation := to mean “defined as”.

By the lemma, there is a ξ(1) ∈ Σ_{1} such that M_{1} := M_{0} ∩ M_{1,ξ(1)} ⊈ j_{L}(U). Since M is closed under finite meets, it follows that M_{1} ∈ M . Another application of the lemma allows us to find a ξ(2) ∈ Σ_{2} such that M_{2} := M_{1} ∩ M_{2,ξ(2)} ⊈ j_{L}(U). Since no term in the descending chain

M_{0} ⊇ M_{1} ⊇ ⋯ ⊇ M_{n} ⊇ ⋯

is included in j_{L}(U), it follows from Proposition 2.5 that ⋂ M_{n} ⊈ j_{L}(U). Since M_{n} ⊆ M_{n,ξ(n)} it also follows that M_{0} ∩ A_{ξ} = M_{0} ∩ ⋂_{n∈N} M_{n,ξ(n)} ⊈ U and hence that M_{0} ∩ A ⊈ U which means that M_{0} ⊈ j_{A}(U), contrary to our assumption.

4.3. Connection with Isbell’s theorem. As mentioned in the introduction, Isbell showed that a locally compact locale and an intersection of a descending sequence of locally compact locales are spatial. This can be used to give a quick proof of Theorem 4.1. A coherent space is locally compact. Moreover, since the compact open sets form a base for the topology, it is immediate that every open set is also locally compact. The only thing to be settled is how to replace the sequence of open sets by a descending sequence.

4.4. Proposition. If A and B are subspaces of a space X, then the sup j_{A} ∨ j_{B} in the lattice of nuclei on O(X) is the smallest nucleus containing the composite j_{A}j_{B}.

Proof. Clearly j_{A} ∨ j_{B} is the least nucleus containing the set theoretic union j_{A} ∪ j_{B}. The conclusion follows immediately from j_{A} ∪ j_{B} ⊆ j_{A}j_{B} ⊆ (j_{A} ∪ j_{B})^{2}.

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Since j_{A}j_{B} is expansive and preserves finite inf, it is clear that the nucleus it generates is the transfinite iteration of the powers of j_{A}j_{B}. By the above inequality, it is also the transfinite iteration of the powers of j_{A} ∪ j_{B} even though the result does not obviously preserve finite inf.

4.5. Corollary. If A is open or B is closed, then j_{A∩B} = j_{A} ∨ j_{B}.

Proof. We know that for open U and V we have V ⊆ j_{A∩B}(U) if and only if V ∩A∩B ⊆ U. If A is open then (V ∩A)∩B ⊆ U if and only if V ∩A ⊆ j_{B}(U) if and only if V ⊆ j_{A}j_{B}(U). If B is closed, then V ∩ A ∩ B ⊆ U if and only if V ∩ A ⊆ ¬B ∪ U = j_{B}(U) if and only if V ⊆ j_{A}j_{B}(U). In either case we see that j_{A∩B} = j_{A}j_{B} whence j_{A}j_{B} is already a nucleus and therefore is j_{A} ∨ j_{B}.

We can now complete the second proof of Theorem 4.1. For if {U_{n}} is a sequence of open sets in X, we can replace them by U_{1}, U_{1} ∩ U_{2}, U_{1} ∩ U_{2} ∩ U_{3}, … to get a descending sequence. As already noted, these will all be locally compact when X is coherent.

4.6. Extension to locally closed sets. Joyal also raised the question of extending the result of Theorem 4.1 to an intersection of locally closed sets. We will see that this is correct.

4.7. Proposition. An intersection of closed sets is spatial.

Proof. Suppose that F_{α} is a family of closed sets and F = ⋂ F_{α}. If j is any nucleus for which j ≥ j_{F_{α}} for all α, then for any α and any open U we have j(U) ⊇ ¬F_{α} ∪ U so that j(U) ⊇ ⋃(¬F_{α}) = ¬F ∪ U = j_{F} (U) so that ⋁ j_{F_{α}} ≥ j_{F} while the reverse inequality is evident.

4.8. Corollary. In any space in which every localic inf of every sequence of opens is spatial, the same is true of every localic inf of locally closed subsets.

Proof. Let {B_{n}} be such a sequence. Write B_{n} = U_{n} ∩ F_{n} with U_{n} open and F_{n} closed. Let A = ⋂ U_{n} and F = ⋂ F_{n}. Since ⋀ B_{n} = ⋀ U_{n} ∧⋀ F_{n} = A∧F = A∩F, the conclusion follows.

5. Connections with a Theorem of Makkai’s

Theorem 4.1 can be derived from an unpublished theorem of Michael Makkai’s that extends a famous result of [Rasiowa & Sikorski, 1950], which can also be found in [Rasiowa & Sikorski, 1968, p. 88]. In order to discuss the connection, we need to recall a few basic concepts. If L is a locale, then a point of L is defined to be a localic map p : 1 [figure] L where 1 stands for the one-point topological space (regarded as a locale). In other words, a point of L is a frame homomorphism p : L [figure] {⊥, ⊤}. For example, if L = O(X) is a spatial locale, then every element x ∈ X determines a point x̂ for which x̂(U) = ⊤ if and only if x ∈ U.

5.1. Enough points We say that the locale L has enough points if whenever x ≠ y there is a point p : L [figure] {⊥, ⊤} for which p(u) ≠ p(v). Recall that a locale is spatial if and only if it is isomorphic to the locale of all open subsets of a topological space. The following straightforward result is well-known:

5.2. Proposition. A locale is spatial if and only if it has enough points.

Proof. If L is isomorphic to O(X) where X is a topological space, then it clearly has enough points of the form x̂ for x ∈ X.

Conversely, assume that L has enough points. Let pt(L) be the set of all points of L and for each u ∈ L define an open subset û ⊆ pt(L) by û = {p ∈ X | p(u) = ⊤}. It readily follows that L is isomorphic to O(pt(L)).

In the next proposition, we use the obvious fact that a T_{0} space is sober if and only every indecomposable closed set has at least one generic point.

5.3. Proposition. The set theoretic meet of any family of sober subspaces of a T_{0} topological space is sober.

Proof. Let X be a T_{0} space and let {Y_{α}} be a family of sober subspaces of X. Let p : 1 [figure] Y be any point of Y = ⋂ Y_{α}. Then, for each α, there is a corresponding point p_{α} : 1 [figure] Y_{α} given by p_{α} = i_{α}p where i_{α} is the inclusion Y ⊆ Y_{α}. Since Y_{α} is sober, the point p_{α} is represented by an element x_{α} ∈ Y_{α}. By factoring through the inclusion of Y_{α} [figure] X we get a point of X which is represented by x_{α}. Since X is T_{0}, the elements x_{α} must all coincide, and must therefore be in Y.

5.4. Proposition. Let {Y_{α}} be a family of sober subspaces of a T_{0} topological space X. Then, Y := ⋂ Y_{α}, the intersection of the family in the lattice of all subspaces coincides with L := ⋀ Y_{α}, the intersection in the lattice of all sublocales of X, if and only if the sublocale ⋀ Y_{α} has enough points.

Proof. If the two intersections coincide, then L is spatial so it must have enough points. Conversely, assume that L has enough points. Since L is contained in each spatial sublocale Y_{α}, we see that every point of L is a point of each Y_{α} which, by sobriety, corresponds to an element of Y_{α} and hence to an element of Y. Let E_{Y_{α}}, E_{Y} and E_{L} denote the congruences on O(X) determined by Y_{α}, Y, and L respectively. Since E_{L} is the sup in the lattice of congruences of the E_{Y_{α}} and E_{Y_{α}} ⊆ E_{Y} for all α, it is immediate that E_{L} ⊆ E_{Y}. Thus we have a surjection O(X)/ E_{L} [figure] O(X)/ E_{Y}. But since the set P of points of L coincides with the set |Y | we have that the bottom row of

[figure: 2^{P}; 2^{|Y |}; ≅; O(X)/ E_{L}; 2^{P}; O(X)/ E_{L}; O(X)/ E_{Y}; O(X)/ E_{Y}; 2^{|Y |}]

is an isomorphism from which it is evident that the top map is also an isomorphism.

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There are lots of sober subspaces in view of the above proposition and:

5.5. Proposition.

1. Every closed subset of a sober space is sober.

2. Every open subset of a sober space is sober.

Proof.

1. Straightforward, by looking at closed indecomposable subsets.

2. Let X be a sober space and let U ⊆ X be an open subset. Let A_{0} ≠ ∅ be a (relatively) closed indecomposable subset of U. Let A be the closure of A_{0} in X. We first claim that A is indecomposable in X. Suppose A = B ∪C. Let B_{0} = B ∩U and C_{0} = C ∩ U. Then A_{0} = B_{0} ∪ C_{0}. Since A_{0} is indecomposable in U we see that either A_{0} = B_{0} or A_{0} = C_{0}. Say A_{0} = B_{0}. Then the closure of B_{0} is contained in B but the closure of B_{0} is the closure of A_{0}, which is A so A = B.

Now, since A is indecomposable in X there exists an element x ∈ X such that A is the closure of {x}. It suffices to show that x ∈ U. Let u ∈ A_{0}. Then u ∈ A and therefore is in the closure of {x}. Thus x is in every neighbourhood of u. Since U is a neighbourhood of u we must have x ∈ U.

5.6. Proposition. Let X be a topological space and let {A_{α}} be a family of closed subsets of X. Then ⋂ A_{α}, the intersection in the lattice of all subspaces of X, coincides with ⋀ A_{α}, the intersection in the lattice of all sublocales of X.

Proof. For each α, let W_{α} ⊆ X be the complement of A_{α}. Then each W_{α} is obviously open. Let j_{α} be the nucleus of A_{α}, viewed as a sublocale of O(X). It is readily shown that j_{α} is given by j_{α}(U) = U ∨ W_{α} for all U ∈ O(X). Now let j be the sup of {j_{α}} in the lattice of all nuclei on O(X). It is readily shown that j(U) = U ∨ (⋁ W_{α}) but this is the nucleus for ⋂ A_{α}, viewed as a sublocale of O(X).

5.7. Makkai’s Theorem. The theorem of Rasiowa and Sikorski mentioned at the beginning of this section can be paraphrased as follows.

5.8. Theorem. Let A be a Boolean algebra and Q be a countable family of subsets of A. Let B be the Boolean algebra freely generated by A together with one element forced to be a sup for each set in Q. Then there are enough 2-valued Boolean representations of B that preserve all the sups from Q to separate the points of A.

Had the conclusion been that there were enough such “points” to separate the points of B, this would have given a different proof of our Theorem 4.1 in the special case of a Stone space. However, in a so-far unpublished work, Makkai has strengthened the Rasiowa-Sikorski theorem in two ways: the theorem is generalized to meet semi-lattices and the conclusion has been strengthened in the way required to give an alternate proof of our theorem in the general case.

5.9. Theorem. [Makkai, unpublished] Assume that P is a meet-semi-lattice with a coverage system generated by Y_{1} ∪ Y_{2} where Y_{1} is a countable set of covers and Y_{2} is a set of finite covers. Then the locale generated by these data (see [Johnstone, 1982, pp. 57–59]) has enough points.

We now sketch how this result can be used to give an alternate proof of 4.1.

Proof. We work in the meet-semilattice M of all compact open subsets of X. For each i, we write U_{i} = ⋃ Σ_{i} where Σ_{i} ⊆ M . We let Y_{1} be the countable set of covers given by saying that Σ_{i} is a cover of the top element of O(X). We let Y_{2} be all covers of the form (M, C) where M ∈ M and C ⊆ M is a finite subset for which ⋃ C = M. It is readily shown that the locale generated by the meet-semilattice M with the coverage system generated by Y_{1} ∪ Y_{2} is the sublocale ⋀ U_{i}. By Makkai’s result, it follows that ⋀ U_{i} has enough points, and the proof of this proposition then follows from 5.4.

6. Applications to the Boolean cyclic spectrum

In this section, we use the result about countable meets of open subsets of a coherent space to obtain results about the cyclic spectrum of a Boolean flow. We briefly review the earlier papers, [Kennison, 2002], [Kennison, 2006], explaining what a Boolean flow is, and why we want to study it. We also review our notation (some of which has been changed to make it more descriptive).

(Discrete) Dynamical Systems. Conceptually, imagine that we have a “system” that can be in different states. Let X be the set of all possible states and assume that X has a topology. Imagine further that, when in state x, the system will, after a fixed period of time, make a transition to a state t(x). We assume that t : X [figure] X is continuous. Then if the system starts out in state x its future states will form a sequence {x, t(x), t^{2}(x), …, t^{n}(x), …} called the orbit of x. Such an orbit might eventually cycle or approach a cycle or behave chaotically and so forth. We want to break X down into components which are “close” to being cyclic.

Formally, a flow in a category C is a pair (X, t) where X is an object of C and t : X [figure] X is a map of C. If (X, t) and (Y, s) are flows in C then a map h : X [figure] Y is a flow homomorphism if ht = sh. This enables us to define Flow(C), the category of flows in C. In view of the above comments, we wish to study the category of flows on topological spaces. If (X, t) is such a flow, it is fruitful to assume that X is compact and Hausdorff but further assuming that X is totally disconnected, and therefore a Stone space, is unnecessary. However, the well-known method of symbolic dynamics can be used to approximate a flow on any topological space by a flow on a Stone space. Roughly speaking, given a flow on a space X, we write X as a finite union:

X = ⋃{A_{σ} | σ ∈ Σ}

where each A_{σ} is a closed subset and Σ is a finite set of “symbols”. Then each x ∈ X is “compatible” with at least one sequence (σ_{0}, σ_{1}, …, σ_{n}, …) meaning that t^{n}(x) ∈ A_{σ_{n}},

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for all n ∈ N. (Since the subsets {A_{σ}} may overlap, each x may be compatible with more than one such sequence of symbols. In practice, the family {A_{σ}} of subsets is chosen to make the overlaps as small as possible.) The compatible sequences (i.e. sequences {σ_{n}} for which there exists x ∈ X compatible with {σ_{n}}) form a subset of Σ^{N} which is closed under the “truncation map” tr : Σ^{N} [figure] Σ^{N} where:

tr(σ_{0}, σ_{1}, σ_{2}, …) = (σ_{1}, σ_{2}, …)

(Note that if x is compatible with the sequence s then t(x) is compatible with the sequence tr(s).) If X̂ is the closure of the set of all compatible sequences in Σ^{N} then (X̂, tr) is a flow in Stone spaces which approximates the original flow (X, t). In what follows, we will assume that (X, t) is a flow on a Stone space X. We will also, in some of our examples, denote the truncation map by t instead of tr and not use the notation X̂. For details, see [Kennison, 2006].

The Boolean Cyclic Spectrum. In view of the above, we will study flows on Stone spaces. By the Stone Duality Theorem, it is obvious that the category of flows on Stone spaces is dual to the category of flows on Boolean algebras. (Given as flow (X, t) on a Stone space X, we let (B, τ) denote the corresponding flow on the Boolean algebra B of all clopen subsets of X where τ = t^{−1}).

We define when a Boolean flow (B, τ) is cyclic (see below) and this definition readily extends to what is meant by a cyclic sheaf of Boolean flows over a locale. The cyclic spectrum of (B, τ) is then the “best” way to map B to the global sections of a cyclic sheaf of Boolean flows over a locale (see below). This is analogous to the local ring sheaf of a ring, see [Kennison, 2006] and [Johnstone, 2002] for details. We outline the main construction, our notation and the main results of the previous papers

1. We let (B, τ) denote a flow in Boolean algebras and let (X, t) be the corresponding flow on Stone spaces, where B is the Boolean algebra of clopens of X.

2. We say that as Boolean flow (B, τ) is cyclic if for all b ∈ B there exists n > 0 such that τ^{n}(b) = b.

3. We let Ẑ denote the profinite integers, which is the inverse limit of the groups {Z_{n} | n ≥ 2} together with the obvious quotient maps Z_{m} [figure] Z_{n} whenever m is a multiple of n. The Boolean flow (B, τ) is cyclic if and only if there is a compatible action α : Ẑ×B [figure] B such that, for all n > 0 and all b ∈ B, we have α(n, b) = t^{n}(b). This, in turn, is true if and only if there is a similar compatible action α : Ẑ×X [figure] X where X is the Stone space corresponding to B.

4. We say that I ⊆ B is a “flow ideal” of (B τ) if I is an ideal such that there is a homomorphism B/I [figure] B/I which commutes with τ : B [figure] B. It readily follows that a flow ideal is a Boolean ideal I ⊆ B satisfying the condition that whenever b ∈ I then τ(b) ∈ I.

5. We let W be the space of all flow ideals of (B, τ). For each b ∈ B, we define

N(b) ⊆ W by:

N(b) = {I ∈ W | b ∈ I}

We topologize W so that the family {N(b)} forms a base. We let O(W) denote the locale of all open subsets of W. By the argument given in Example 3.7, the space W is coherent.

6. There is a canonical sheaf Q of Boolean flows over W where the stalk Q_{I} is defined as B/I for each I ∈ W. (Note: In the previous papers, the sheaf Q was denoted by B^{0}. Here we use Q, which is, in a sense, the universal quotient flow of (B, τ), see [Kennison, 2006]).

7. We let O(W)_{cyc} be the largest sublocale of O(W) which forces Q to become cyclic. That is, O(W)_{cyc} is the largest sublocale such that, for each b ∈ B, the family {N(b − τ^{k}b)} covers O(W)_{cyc}. The sheaf Q restricts to a sheaf Q_{cyc} on the locale O(W)_{cyc} and Q_{cyc} is a cyclic Boolean flow over the locale O(W)_{cyc}. It is the “best” such sheaf and is called the Boolean cyclic spectrum of (B, τ). Previously, O(W)_{cyc} and Q_{cyc} were denoted by L_{cyc} and B^{∗} respectively.

8. A flow ideal I ∈ W is a cyclic flow ideal if the Boolean flow B/I is cyclic. We let W_{sp.cyc} ⊆ W denote the subspace of all cyclic flow ideals. It is shown in [Kennison, 2006] that O(W)_{cyc} is spatial (isomorphic to the locale of opens of a topological space) if and only if it is isomorphic to O(W_{sp.cyc}). But in [Kennison, 2006], L_{cyc} is used for O(W)_{cyc} and W_{cyc} is used for W_{sp.cyc}. To avoid confusion, the notations L_{cyc} and W_{cyc} will not be used in this paper.

9. We let Γ(Q_{cyc}) denote the set of global sections over the cyclic spectrum.

10. For further details, consult the previous papers, particularly [Kennison, 2006].

In the previous papers, we proved the results involving the actions by Ẑ and proved that Q and Q_{cyc} have the universal properties mentioned above. We described Q and Q_{cyc} in the case that (B, τ) is finitely generated (as a flow). We also pursued the questions of whether O(W)_{cyc} is spatial and of how to describe the global sections over the cyclic spectrum. In this paper, using Theorem 4.1, we will:

1. show that the cyclic spectrum of a countable flow is always spatial;

2. give an explicit description of the nucleus j_{cyc};

3. show that the cyclic spectrum always has the Lindelöf property;

4. describe Γ(Q_{cyc}), the set of global sections over the cyclic spectrum.

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The cyclic spectrum of a countable flow is spatial. The following extends the result in [Kennison, 2006] that the cyclic spectrum of a finitely generated Boolean flow is spatial. Example 7.1 shows that for uncountable B, the cyclic spectrum need not be spatial, which answers a question left open in [Kennison, 2006].

6.1. Proposition. If B is countable then the locale O(W)_{cyc} is spatial (so we can regard Q_{cyc} as a sheaf over the space W_{sp.cyc} and we say that B has a spatial cyclic spectrum).

Proof. The base of the cyclic spectrum, O(W)_{cyc}, can be defined as the largest sublocale of O(W) for which every b ∈ B becomes cyclic, meaning that, for each such b, the basic open sets {N(b − τ^{k}b)} cover O(W)_{cyc}. Note that N(b) is defined above; for more details, see [Kennison, 2006]. It follows that, if we let cyc(b) = ⋃{N(b − τ^{k}b) | k > 0}, then O(W)_{cyc} is the localic meet of {cyc(b) | b ∈ B}. If B is countable, then this meet is spatial by Theorem 4.1.

For technical reasons, we want to generalize the above result. We need the following definition.

6.2. Definition. Let C ⊆ B be a countable subset. Let W_{C} be the largest sublocale of O(W) which makes every c ∈ C cyclic. That is, W_{C} is the localic meet of {cyc(c) | c ∈ C}. Furthermore, we say that a flow ideal I ∈ W is C-cyclic if for every c ∈ C, there exists k > 0 such that I ∈ N(c − τ^{k}c).

6.3. Proposition. The sublocale W_{C} is spatial for every countable C ⊆ B. The sublocale W_{C} ⊆ W can be identified with the subspace of all C-cyclic flow ideals.

Proof. The proof of the previous proposition clearly applies here.

6.4. Remark. We will routinely identify W_{C} with the subspace of all C-cyclic flow ideals.

Description of the nucleus j_{cyc} and the Lindelöf property. Our next result gives a fairly technical, but quite useful, characterization of j_{cyc}. We first need some definitions and notation.

6.5. Definition. An open set U ∈ O(W) is countably basic if we can write U as a countable union of basic open subsets of the form N(b) for b ∈ B.

6.6. Theorem. Let (B, τ) be a Boolean flow and let b ∈ B and U ∈ O(W) be given. Then N(b) ⊆ j_{cyc}(U) if and only if there exists a countable subset C ⊆ B and a countably basic open set U_{0} ⊆ U such that N(b) ∩ W_{C} ⊆ U_{0}.

Proof. We define J(U) as the union of all N(b) for which there exists a countable subset C ⊆ B and a countably basic open set U_{0} ⊆ U such that N(b) ∩ W_{C} ⊆ U_{0}. We claim that J is a nucleus. The only non-trivial step is proving that J is idempotent. By examining the nucleus j_{C} for the subspace W_{C} ⊆ W, it readily follows that N(b) ∩ W_{C} ⊆ U_{0} if and only if N(b) ⊆ j_{C}(U_{0}). Assume N(b) ⊆ J(J(U)). Then there exists a countable subset C ⊆ B and a countably basic set V ⊆ J(U) such that N(b) ⊆ j_{C}(V). Write V = ⋃ N(d_{n}) where N(d_{n}) ⊆ J(U) for all n ∈ N. Then for each n, there exists a countably basic V_{n} ⊆ U and a countable subset C(n) ⊆ B with N(d_{n}) ⊆ j_{C(n)}(V_{n}). Let U_{0} = ⋃ V_{n} and D = C ∪ ⋃ C(n). It suffices to show that N(b) ⊆ j_{D}(U_{0}). But j_{D} ≥ j_{C} and j_{D} ≥ j_{C(n)} for all n. So V ⊆ j_{D}(U_{0}) and j_{C}(V) ⊆ j_{D}(j_{D}(U_{0})) = j_{D}(U_{0}) and the claim follows.

The nucleus J makes every b ∈ B cyclic (let C = {b} then U_{0} = ⋃ N(b − τ^{n}b) covers W_{C} and is countably basic). It follows that J ≥ j_{cyc} and the opposite inclusion, J ≤ j_{cyc} is obvious.

6.7. Definition. A locale L has the Lindelöf property if whenever F ⊆ L covers L (that is whenever ⋁ F = ⊤) then F has a countable subset F_{0} ⊆ F which also covers L.

6.8. Proposition. The locale O(W)_{cyc} has the Lindelöf property.

Proof. It suffices to show that any cover of O(W)_{cyc} by basic opens N(b) has a countable subcover. Suppose that U = ⋃ N(b_{α}) and that U covers O(W)_{cyc}. Then j_{cyc}(U) = ⊤ = N(0) so, by the above theorem, there is a countably basic U_{0} ⊆ U with j_{cyc}(U_{0}) = ⊤. Let U_{0} = ⋃ N(c_{n}). Then for each n we have N(c_{n}) ⊆ ⋃ N(b_{α}) which readily implies that there exists α with N(c_{n}) ⊆ N(b_{α}) and so only a countable set of the N(b_{α}) is needed to cover U_{0} and hence to cover O(W)_{cyc}.

Description of the global sections over the cyclic spectrum. It remains to describe the Boolean flow Γ(Q_{cyc}) of all global sections over the cyclic spectrum of B. We first do this when B is countable then show how to extend that result to arbitrary B.

We will use the following convenient assumption for the rest of this section.

6.9. Assumption. Let (B, τ) be a Boolean flow. By Stone duality, we can suppose that B = clop(X), the algebra of clopen sets of the Stone space X. Another use of Stone duality shows that there is a unique continuous map t : X [figure] X such that τ(b) = t^{−1}(b) for all b ∈ B. Note that we are not merely assuming that B is isomorphic to clop(X) but that every b ∈ B is actually a clopen set of X . See [Kennison, 2002] for more details.

6.10. Definition. Let X be as above, let b ∈ B be a clopen subset of X and let x ∈ X be given. We say that x is k-cyclic with respect to b if, for all n ≥ 0, we have t^{n}(x) ∈ b if and only if t^{n+k}(x) ∈ b.

We say that x is cyclic with respect to b if x is k-cyclic with respect to b for some k > 0 (in this case, k is a period of x).

Further, x ∈ X is cyclic if, for all b ∈ B, x is cyclic with respect to b. We let X_{cyc} denote the subspace of all cyclic elements of X.

We let k-Cy(b) denote the set of all x ∈ X which are k-cyclic with respect to b.

6.11. Examples. Suppose (B, τ) is a cyclic Boolean flow, meaning that for every b ∈ B there exists k > 0 such that b = τ^{k}b. Then X = X_{cyc}.

Another example is given by (X, t) where X = {0, 1}^{N} and t is the truncation map. Then x ∈ X_{cyc} if and only if x is a periodic sequence. See Example 7.5 and the definition that precedes it for details.

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6.12. Definition. Let I ⊆ B be a flow ideal. Then I corresponds to the flow quotient B/I which, by Stone duality, corresponds to a closed subflow A(I) ⊆ X.

If b ∈ B, we let ⟨b⟩ denote the flow ideal generated by b. By abuse of language, we use A(b) to denote A(⟨b⟩).

6.13. Lemma.

1. Let I ⊆ B be a flow ideal. Then A(I) = ⋂{¬b | b ∈ I}. Also, b ∈ I if and only if b ∩ A(I) = ∅.

2. Let I, J ⊆ B be flow ideals. Then I ⊆ J if and only if A(J) ⊆ A(I).

3. Let b ∈ B be given and regard b as a clopen subset of X. Then x ∈ A(b) if and only if t^{n}(x) ∉ b for all n ≥ 0.

Proof.

1. First, we show that if A ⊆ X is a closed subflow, then the corresponding flow ideal is {b | b∩A = ∅}. Let i : A [figure] X be the inclusion map. Then i^{−1} : clop(X) [figure] clop(A) is the corresponding quotient of B = clop(X). Obviously i^{−1}(b) = 0 if and only if A ∩ b = ∅.

It follows that if A is the closed subflow that corresponds to the flow ideal I, then A(I) ⊆ ⋂{¬b | b ∈ I}. It is readily checked that ⋂{¬b | b ∈ I} is topologically closed and closed under the action of t (as I is closed under the action of τ). Suppose d ∩ ⋂{¬b | b ∈ I} = ∅. We must show that d ∈ I. It follows that d is covered by the elements of I and, by compactness, by a finite subset of I. Since I is closed under finite unions, there exists b ∈ I with d ≤ b and this implies that d ∈ I.

2. Straightforward, in view of the first paragraph, above.

3. Clearly A(b) is the largest closed subflow of X which is disjoint from b. A straightforward check shows that the given description of A(b) has this property.

6.14. Lemma. Let b ∈ B and k > 0 be given. Then k-Cy(b) = A(b − τ^{k}b).

Proof. Straightforward.

6.15. Lemma. Let b ∈ B and k > 0 be given. Then for every non-zero multiple m of k, we have ⟨b − τ^{m}b⟩ ⊆ ⟨b − τ^{k}b⟩.

Proof. It clearly suffices to show that if I ⊆ B is a flow ideal and (b − τ^{k}b) ∈ I then (b − τ^{m}b) ∈ I. But suppose (b − τ^{k}b) ∈ I. By applying τ^{k}, we see that (τ^{k}b − τ^{2k}b) ∈ I. Adding (b − τ^{k}b) to it gives us (b − τ^{2k}b) ∈ I and the result follows by an easy induction.

6.16. Proposition. For I ∈ W, we have I ∈ W_{sp.cyc} if and only if A(I) ⊆ X_{cyc}.

Proof. Assume I ∈ W_{sp.cyc} and let x ∈ A(I) be given. To prove that x ∈ X_{cyc}, suppose b ∈ B. Since I is a cyclic flow ideal, there exists k > 0 such that (b − τ^{k}b) ∈ I. It readily follows that A(I) ⊆ A(b − τ^{k}b) and, in view of lemma 6.14, we see that x is k-cyclic with respect to b. Since b is an arbitrary member of B, we see that x ∈ X_{cyc}.

Conversely, assume A(I) ⊆ X_{cyc} and that b ∈ B is given. It easily follows from Lemma 6.14 that {¬(b − τ^{k}b) | k > 0} covers X_{cyc}. Since A(I) ⊆ X_{cyc} it is covered by a finite set {¬(b − τ^{k(i)}b)}. Let m be a common multiple of the set {k(i)}, then it is readily shown that A(I) ⊆ ¬(b − τ^{m}b) which shows that (b − τ^{m}b) ∈ I.

6.17. Proposition. Assume that B is countable (or more generally, that B has a spatial cyclic spectrum). Let d ∈ B be given. Let d̂ denote the corresponding constant section in Γ(Q) and let d̂_{cyc} denote the restriction of d̂ to the subspace W_{sp.cyc}. Then d̂_{cyc} = 0 if and only if d ∩ X_{cyc} = ∅.

Proof. Recall that d is a clopen subset of X. Assume that d ∩ X_{cyc} = ∅. Let I ∈ W_{sp.cyc} be given. As shown above, A(I) ⊆ X_{cyc} so d ∩ A(I) = ∅ and therefore d ∈ I. Since d ∈ I for all I ∈ W_{sp.cyc}, it follows that d̂_{cyc}, the restriction of d̂ to W_{sp.cyc} is 0.

Conversely, assume that d̂_{cyc} = 0 and that x ∈ d ∩ X_{cyc}. We need to derive a contradiction. Since x ∈ X_{cyc}, we can, for every b ∈ B, find a positive integer k(b) such that x is k(b)-cyclic with respect to b. This implies that t^{n}(x) ∉ (b − τ^{k(b)}b) for all n ≥ 0. Let I be the set of all c ∈ B such that t^{n}(x) ∉ c for all n ≥ 0. Then I is readily seen to be a flow ideal of B and a cyclic flow ideal as (b − τ^{k(b)}b) ∈ I for all b ∈ B. Moreover, x ∈ A(I) so d ∩ A(I) ≠ ∅ as x ∈ d ∩ A(I). So d ∉ I and this implies that d̂(I) ≠ 0 which contradicts the assumption that d̂_{cyc} = 0.

6.18. Proposition. Let c, d ∈ B be given (and regard each element of B as a clopen subset of X). Then

N(c) ∩ W_{sp.cyc} ⊆ N(d) if and only if A(c) ∩ X_{cyc} ⊆ A(d)

Proof. First, assume A(c) ∩ X_{cyc} ⊆ A(d). Let I ∈ N(c) ∩ W_{sp.cyc} be given. We need to show that d ∈ I. Since c ∈ I, we have ⟨c⟩ ⊆ I so A(I) ⊆ A(c). By Proposition 6.16, we have A(I) ⊆ X_{cyc}, so, by our assumption, A(I) ⊆ A(d). But then, by Lemma 6.13, (2), ⟨d⟩ ⊆ I and d ∈ I.

Conversely, assume N(c) ∩ W_{sp.cyc} ⊆ N(d). Let x ∈ A(c) ∩ X_{cyc} be given. Since x ∈ X_{cyc}, we can choose, for each b ∈ B, an integer k(b) > 0 such that x is k(b)-cyclic with respect to b. Let I be the flow ideal generated by c and {b−τ^{k(b)}b | b ∈ B}. Then I is the smallest flow ideal containing c and each b−τ^{k(b)}b so A(I) is the largest closed subflow contained in A(c) and each A(b − τ^{k(b)}b) which means that A(I) = A(c) ∩ ⋂_{b} A(b − τ^{k(b)}b). By the choice of k(b), we have x ∈ ⋂_{b} A(b − τ^{k(b)}b) and we assumed that x ∈ A(c) so x ∈ A(I). Clearly t^{n}(x) ∈ A(I) for all n ≥ 0. But by our assumption that N(c)∩W_{sp.cyc} ⊆ N(d), we see that d ∈ I so d ∩ A(I) = ∅ so t^{n}(x) ∉ d (as t^{n}(x) ∈ A(I)) which implies that x ∈ A(d).

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6.19. Corollary. Let c_{1}, c_{2}, d ∈ B be given. Then:

N(c_{1}) ∩ N(c_{2}) ∩ W_{sp.cyc} ⊆ N(d) if and only if A(c_{1}) ∩ A(c_{2}) ∩ X_{cyc} ⊆ A(d)

Proof. This follows from the above proposition with c = c_{1} ∨ c_{2}. Note that N(c_{1} ∨ c_{2}) = N(c_{1}) ∩ N(c_{2}) and A(c_{1} ∨ c_{2}) = A(c_{1}) ∩ A(c_{2}).

6.20. Definition. A subset S ⊆ X_{cyc} is rectified by b ∈ B if there exists d ∈ B such that

S ∩ A(b) = d ∩ A(b) ∩ X_{cyc}

We let Rect(S) denote the set of all b ∈ B which rectify S. We say that S ⊆ X_{cyc} is regular if

W_{sp.cyc} ⊆ ⋃{N(b) | b ∈ Rect(S)}.

6.21. Proposition.

1. If d ∈ B, then d ∩ X_{cyc} is a regular subset of X_{cyc}.

2. The regular subsets of X_{cyc} are closed under complementation (within X_{cyc}) and under finite unions and intersections (which includes the empty subset and X_{cyc} itself).

Proof.

1. First, it is clear that every b ∈ B rectifies d ∩ X_{cyc} and when b = ⊥, then N(b) covers W.

2. Closure under complementation follows by verifying that b rectifies S if and only if b rectifies X_{cyc} − S. To prove closure under pairwise intersections, it suffices to verify that if b ∈ Rect(S) and c ∈ Rect(T), then b ∨ c ∈ Rect(S ∩ T). Note that if S is empty, then every b ∈ B rectifies S. The rest of the proof follows by considering complements within X_{cyc}.

6.22. Notation. Assume that B is countable (or more generally, that B has a spatial cyclic spectrum). We let Reg(B) denote the Boolean algebra of all regular subsets of X_{cyc}. In view of (1) of the above proposition, there is a canonical Boolean homomorphism from B to Reg(B).

6.23. Lemma. If d, e ∈ B are given, then d ∩ A(d − e) ⊆ e.

Proof. Assume the contrary, that there exists x ∈ d ∩ A(d − e) but with x ∉ e. Then x ∈ (d − e) ∩ A(d − e) which is a contradiction.

6.24. Theorem. Assume that B is countable (or more generally, that B has a spatial cyclic spectrum). The Boolean algebra Γ(Q_{cyc}) is canonically isomorphic to Reg(B). Moreover, the isomorphism commutes with the map B [figure] Reg(B) mentioned above and the map B [figure] Γ(Q_{cyc}) which sends d ∈ B to d̂.

Proof. Let σ ∈ Γ(Q_{cyc}) be given. Locally, σ agrees with constant sections of the form d̂ so we can find a family {(d_{α}, b_{α})} such that σ = d̂_{α} on N(b_{α}) ∩ W_{sp.cyc}. It follows that these sections are compatible, meaning that

N(b_{α}) ∩ N(b_{β}) ∩ W_{sp.cyc} ⊆ N(d_{α} − d_{β})

Now define S ⊆ X_{cyc} as ⋃_{α}(d_{α} ∩ A(b_{α}) ∩ X_{cyc}). We claim that each b_{β} rectifies S. We must show that S ∩ A(b_{β}) = d_{β} ∩ A(b_{β}) ∩ X_{cyc}. We have:

S ∩ A(b_{β}) = ⋃_{α}(A(b_{β}) ∩ d_{α} ∩ A(b_{α}) ∩ X_{cyc})

We note that if β = α then (A(b_{β}) ∩ d_{α} ∩ A(b_{α}) ∩ X_{cyc}) reduces to d_{β} ∩ A(b_{β}) ∩ X_{cyc}, so it suffices to show in general that (A(b_{β}) ∩ d_{α} ∩ A(b_{α}) ∩ X_{cyc}) ⊆ d_{β}. By Corollary 6.19, and the above condition that N(b_{α}) ∩ N(b_{β}) ∩ W_{sp.cyc} ⊆ N(d_{α} − d_{β}), we see that

A(b_{α}) ∩ A(b_{β}) ∩ X_{cyc} ⊆ A(d_{α} − d_{β}).

So (A(b_{β}) ∩ d_{α} ∩ A(b_{α}) ∩ X_{cyc}) ⊆ d_{α} ∩ A(d_{α} − d_{β}). The claim now follows by the above lemma. The claim implies that S is regular, so we have associated the regular set S to the global section σ

Conversely, let S ∈ Reg(B) be given. Let {b_{α}} be a family of elements of B which rectify S and cover W_{sp.cyc}. Then for each α there exists d_{α} such that

S ∩ A(b_{α}) = d_{α} ∩ A(b_{α}) ∩ X_{cyc}.

Observe that for all α, β:

d_{α} ∩ A(b_{α}) ∩ A(b_{β}) ∩ X_{cyc} = d_{β} ∩ A(b_{α}) ∩ A(b_{β}) ∩ X_{cyc}

as both are S ∩ A(b_{α}) ∩ A(b_{β}). Since A(b_{α}) ∩ A(b_{β}) ∩ X_{cyc} is a subflow (closed under the action of t) the above result readily implies that

A(b_{α}) ∩ A(b_{β}) ∩ X_{cyc} ⊆ A(d_{α} − d_{β}).

And by Corollary 6.19, this implies that

N(b_{α}) ∩ N(b_{β}) ∩ W_{sp.cyc} ⊆ N(d_{α} − d_{β}).

But this is precisely what we need to show that the local sections d̂_{α} on N(b_{α}) piece together to give us a global section σ.

So, to each global section σ we have associated a regular set S and to each regular set S we have associated a global section σ. A routine check shows that this defines the desired isomorphism.

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Global sections of Γ(Q_{cyc}) when B need not be countable. We recall the definition of W_{C} for each countable subset C ⊆ B. As noted above, W_{C} is a spatial locale for each such countable subset C. Let Q_{C} be the restriction of Q, the canonical sheaf over W, to the subspace W_{C}. The global sections Γ(Q_{C}) can be determined by an approach strictly similar to the approach in the above theorem. That is, we can define X_{C} ⊆ X, as the set of all x ∈ X which are cyclic with respect to every c ∈ C. We can then define a subset of X_{C} to be C-regular by an obvious modification of the definition of regular (in fact, just replace X_{cyc} by X_{C}). The argument used in the proof of 6.24 can then be used to show that Γ(Q_{C}) is canonically isomorphic to the family of all C-regular subsets of X_{C}. Then the global sections over the cyclic spectrum, for arbitrary B, can be described using the following theorem.

6.25. Theorem. Γ(Q_{cyc}) is the colimit of Γ(Q|W_{C}) where C varies over the filtered family of all countable subsets of B.

Proof. We must prove that every global section in Γ(Q_{cyc}) is the restriction of a global section in Γ(Q|W_{C}) for some countable subset C ⊆ B. We must also show that two such global sections over W_{C} and W_{D} have the same restriction to O(W)_{cyc} if and only if they have the same restriction to some W_{E} where E ⊆ B is a countable subset with C ∪D ⊆ E.

Clearly, every global section σ ∈ Γ(Q_{cyc}) is represented by a compatible family {(d_{α}, b_{α})} for which σ equals d̂_{α} on N(b_{α}). Since O(W)_{cyc} is Lindelöf, we can assume that the family is countable and write it as {(b_{n}, d_{n}) | n ∈ N}. The condition for the family being compatible is equivalent to a countable set of conditions of the form N(b_{α}) ∩ N(b_{β}) ⊆ j_{cyc}(N(d̂_{α} − d̂_{β})). But by using Theorem 6.6, this condition holds if and only if it holds when we restrict to some W_{C}. It readily follows that {(d_{α}, b_{α})} will be a compatible family that defines a section in Γ(Q_{cyc}) if and only if it is compatible enough to define a section in Γ(Q|W_{C}) for some countable C ⊆ B. The remaining details are now straightforward.

7. Examples

7.1. Example of a non-spatial cyclic spectrum. In constructing this example, it is notationally convenient to introduce, for each n ∈ N, a symbol a_{n} and for each f ∈ N^{N} a symbol h_{f}. We let

G = {a_{n} | n ∈ N} ∪ {h_{f} | f ∈ N^{N}}

We let (B, τ) be the free Boolean flow generated by G.

For each n ∈ N and f ∈ N^{N} we let

U(n, f) = N(τ^{f(n)}a_{n} − a_{n}) ∩ N(τ^{n}h_{f} − h_{f})

We then claim that:

1. the family {U(n, f)} covers W_{sp.cyc}; 2. the above family has no countable subcover;

3. the cyclic spectrum of this flow is not spatial.

Proof.

1. Let I ∈ W_{sp.cyc} be given. Since I is cyclic, we can clearly define f : N [figure] N such that τ^{f(n)}a_{n} − a_{n} ∈ I for all n ∈ N. But, there also must be an n ∈ N for which τ^{n}h_{f} − h_{f} ∈ I and it follows that I ∈ U(n, f).

2. Assume there is a countable subcover. Then we can clearly find a sequence (f_{1}, f_{2}, … f_{n}, …) of functions from N to N such that {U(n, f_{i}) | n, i ∈ N} covers W_{cyc}.

Now define u : N [figure] N such that u(n) > f_{i}(n) whenever i ≤ n. Let v : N^{N} [figure] N be any function for which v(f_{i}) = i and v(f) > 0 for all f. Let I be the flow ideal generated by:

{τ^{u(n)}a_{n} − a_{n} | n ∈ N} ∪ {τ^{v(f)}h_{f} − h_{f} | f ∈ N^{N}}

Then I is obviously cyclic, so there exist n, i ∈ N with I ∈ U(n, f_{i}).

But this implies that τ^{f_{i}(n)}a_{n} −a_{n} ∈ I and so u(n) < f_{i}(n) which implies that i > n. On the other hand, τ^{n}h_{f_{i}} − h_{f_{i}} ∈ I which implies that n ≥ v(f_{i}) so i ≤ n which is a contradiction.

3. The cyclic spectrum cannot be spatial because, as shown in [Kennison, 2006, Proposition 4.1] this implies that it is a sheaf over the space W_{sp.cyc} and, by Proposition 6.8, that W_{sp.cyc} is Lindelöf, which contradicts the above.

7.2. Examples of regular sets.

7.3. Definition. Let (B, τ) be a Boolean flow. We say that G ⊆ B generates B as a flow if no proper subflow of B contains G,

7.4. Definition. We say that x ∈ X is periodic if there exists n > 0, such that t^{n}x = x. We let Per(X) denote the set of all periodic elements of X. We note that Per(X) ⊆ X_{cyc}.

7.5. Example. Let X = {0, 1}^{N} and define t : X [figure] X as the shift map (so that t(x_{0}, x_{1}, x_{2}, …) = (x_{1}, x_{2}, …)). Let (B, τ) be the corresponding flow in Boolean algebras. Then X_{cyc} = Per(X) is the set of all periodic sequences and every subset of X_{cyc} is regular.

Proof. Note that g = π_{0}^{−1}(1) generates B as a flow and that x ∈ X is a periodic sequence if and only if x is cyclic with respect to g. It readily follows that X_{cyc} = Per(X).

Let b_{n} = τ^{n}g − g and let S ⊆ X_{cyc} be any subset. Then we claim that b_{n} rectifies S. It is readily shown that A(b_{n}) is the set of all sequences in X which are n-periodic, which is a finite set. So every subset of A(b_{n}) is relatively clopen and is clearly of the form d ∩ A(b_{n}). It easily follows that b_{n} rectifies any subset S. But the family of all N(b_{n}) clearly covers W_{sp.cyc} so S is regular.

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7.6. Remark. If (B, τ) is finitely generated (as a flow) then X_{cyc} always coincides with Per(X) and every subset of X_{cyc} is regular, as the above argument generalizes.

The following proposition is useful in finding regular sets.

7.7. Proposition. Let (X, t) be a flow in Stone spaces and let (B, τ) be the corresponding flow in Boolean algebras. Recall the definition of k-Cy(−) from 6.10. Let c ∈ B and the positive integer k be given. Then:

1. k-Cy(c) is regular;

2. S = X_{cyc} ∩ ⋂_{n≥0} τ^{n}(c) is regular.

3. S = X_{cyc} ∩ ⋂_{n≥0} τ^{n}(¬c) is regular.

Proof.

1. Let b_{n} = τ^{n}c − c. It is readily shown that A(b_{n}) = n-Cy(c). Let S = k-Cy(c). It follows that S ∩ A(b_{n}) = (k, n)-Cy(c), where (k, n) = gcd(k, n). A straightforward argument proves that the set (k, n)-Cy(c) is relatively clopen in n-Cy(c) (as we only have to restrict the values of t^{i}x for i = 0, 1, …, n−1). A standard argument, using the compactness of A(b_{n}) = n-Cy(c), shows that there is a clopen set d of X such that (k, n)-Cy(c) = d ∩ A(b_{n}) and, from this, it follows that each b_{n} rectifies S. As noted in the previous proof, this shows that S is regular.

2. For each k > 0 we claim that b_{k} = c − τ^{k}c rectifies S. Since A(b_{k}) = k-Cy(c) (by Lemma 6.14), it suffices to find d ∈ B such that S ∩ k-Cy(c) = d ∩ k-Cy(c) ∩ X_{cyc}. But this readily follows for d = c ∩ τc ∩ … ∩ τ^{k−1}c. Finally {N(b_{k})} covers W_{sp.cyc} since every I ∈ W_{sp.cyc} must make c cyclic and so contain b_{k} for some k.

3. Note that (2) implies (3) in view of the substitution of ¬c for c.

7.8. Example. Let Σ_{0} = {σ_{0}, σ_{1}, …, σ_{n}, …} be a sequence of “symbols” and give Σ_{0} the discrete topology. Let Σ = Σ_{0} ∪ {∞} be its one-point compactification. Let X = Σ^{N} and define t : X [figure] X so that t(x_{0}, x_{1}, x_{2}, …) = (x_{1}, x_{2}, …). Let π_{n} : X [figure] Σ denote the nth projection. For each i ∈ N, let g_{i} ∈ B be defined as π_{0}^{−1}(σ_{i}). Let G = {g_{i}}. For this example, we claim that:

1. G generates B as a flow;

2. Per(X) is a proper subset of X_{cyc};

3. there are regular subsets not of the form b ∩ X_{cyc} for b ∈ B (so not every global section over the cyclic spectrum is of the form b̂ for b ∈ B);

4. not every subset of X_{cyc} is regular.

Before proving the above claims, we insert a useful definition and some lemmas.

7.9. Definition. Let (B, τ) be a Boolean flow and let G ⊆ B generate B as a flow. We say that p ∈ B is G-prescriptive if there exists \overline{g} = (g_{1}, …, g_{m}) ∈ G^{m} and an m-tuple \overline{k} = (k_{1}, …, k_{m}) of positive integers such that

p = p(\overline{g}, \overline{k}) = ⋁_{1≤i≤m}(τ^{k_{i}}g_{i} − g_{i})

The following lemma shows that, in a sense, a G-prescriptive element of B has the effect of prescribing a period to an m-tuple of elements of G.

7.10. Lemma. Let p = p(\overline{g}, \overline{k}) be a G-prescriptive element of B. Then A(p) is the set of all x ∈ X which are k_{i}-cyclic with respect to g_{i} for 1 ≤ i ≤ m.

Proof. It is straightforward to show that x is k-cyclic with respect to g if and only if t^{n}x is never in τ^{k}g − g (for any n ∈ N). The proof then follows.

Recall that ⟨c⟩ is the smallest flow ideal of B which contains c. Also b ∈ ⟨c⟩ if and only if b misses A(c). Remember that for b, c ∈ B we have that c, A(c) and b are subsets of X while N(c) and N(b) are subsets of W.

7.11. Lemma. Let b, c ∈ B be given. Then the following are equivalent:

1. N(c) ⊆ N(b);

2. b ∈ ⟨c⟩;

3. A(c) ⊆ A(b).

Proof. (1) [figure](2): If N(c) ⊆ N(b), then ⟨c⟩ ∈ N(c) ⊆ N(b) so b ∈ ⟨c⟩. Conversely, assume b ∈ ⟨c⟩. If I ∈ N(c) then c ∈ I so ⟨c⟩ ⊆ I and b ∈ ⟨c⟩ ⊆ I so I ∈ N(b).

(2)[figure] (3): Assume b ∈ ⟨c⟩. Then ⟨b⟩ ⊆ ⟨c⟩ which, by the duality between flow ideals of B and closed subflows of X, implies that A(c) ⊆ A(b). Conversely, assume A(c) ⊆ A(b). It follows that b misses A(c) so b ∈ ⟨c⟩.

7.12. Corollary. Let S ⊆ X_{cyc} be given and assume that b rectifies S. If b ∈ ⟨c⟩, then c rectifies S.

Proof. By the above lemma, A(c) ⊆ A(b) and the result easily follows.

7.13. Proposition. Let (B, τ) be a countable Boolean flow and let G ⊆ B generate B as a flow. Let S ⊆ X_{cyc} be given and let G-Rect(S) be the set of all G-prescriptive elements that rectify S. Then S is regular if and only if

W_{sp.cyc} ⊆ ⋃{N(p) | p ∈ G-Rect(S)}

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Proof. Assume that S is regular. Then

W_{sp.cyc} ⊆ ⋃{N(b) | b ∈ Rect(S)}.

Let I ∈ W_{sp.cyc} be given. Since I is a cyclic flow ideal, for each g ∈ G, we can choose k(g) > 0 such that τ^{k(g)}g−g ∈ I. Let I_{0} be the smallest flow ideal containing {τ^{k(g)}g−g | g ∈ G}. Since G generates B as a flow, it readily follows that I_{0} is cyclic so there exists b ∈ Rect(S) such that b ∈ I_{0} . But for any element b ∈ I_{0}, there is a finite set F ⊆ G such that b is in the smallest flow ideal containing τ^{k(g)}g − g for all g ∈ F. Write F = {g_{1}, …, g_{m}}, let \overline{g} = (g_{1}, …, g_{m}) and \overline{k} = (k_{1}, …, k_{m}) where k_{i} = k(g_{i}). Let p = p(\overline{g}, \overline{k}). Then by the choice of F, we see that b ∈ ⟨p⟩. By the above Lemma, we have p ∈ G-Rect(S) and p ∈ I_{0} ⊆ I. Since I is an arbitrary member of W_{sp.cyc}, it follows that

W_{sp.cyc} ⊆ ⋃{N(p) | p ∈ G-Rect(S)}

The converse is trivial.

Proof of Example 7.8

1. Note that the clopen set π_{n}^{−1}(σ_{i}) = τ^{n}(g_{i}). If U ⊆ Σ is a clopen neighbourhood of ∞, then F = {i ∈ N | σ_{i} ∈ Σ − U} is finite and π_{n}^{−1}(U) = ⋀_{i∈F} τ^{n}(¬g_{i}). The remaining details are now straightforward.

2. Note that x ∈ X is k-cyclic with respect to g_{i} precisely when there exists k > 0 such that x_{n} = σ_{i} if and only if x_{n+k} = σ_{i}. Let x_{n} = σ_{i} if and only if 2^{i} is the largest power of 2 that divides n. Then x is 2^{i+1}-cyclic with respect to g_{i}. Since G generates B, it follows that x ∈ X_{cyc}. But there is no k such that t^{k}(x) = x as such a k would have to be a multiple of every power of 2. So x ∈ X_{cyc} − Per(X).

3. Choose g ∈ G. The subset 2-Cy(g) is regular, in view of 7.7, but is clearly not of the form b ∩ X_{cyc} for any clopen subset b ⊆ X (as clopen sets can only restrict x_{n} for finitely many n). It follows that the global section corresponding to 2-Cy(g) is not of the form b̂ for any b ∈ B.

4. Let S be the set of all x ∈ X_{cyc} such that x_{n} ≠ σ_{m} for any even m ∈ N. If S is regular then, by Proposition 7.7, there are enough G-prescriptive elements that rectify S. But suppose p is G-prescriptive and that S ∩ A(p) = d ∩ A(p) ∩ X_{cyc} for some d ∈ B. Choose g_{k} for an odd k such that g_{k} is not involved in any part of p or d. Let x ∈ X be the sequence which is constantly σ_{k}. Then x ∈ S ∩ A(p) so x ∈ d ∩ A(p) ∩ X_{cyc}. But we could just as well have chosen k to be even, in which case x is still in d ∩ A(p) ∩ X_{cyc}, but x is not in S, which leads to a contradiction.

532  MICHAEL BARR, JOHN F. KENNISON, AND R. RAPHAEL

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Department of Mathematics and Statistics McGill University, Montreal, QC, H3A 2K6 Department of Mathematics and Computer Science Clark University, Worcester, MA 01610 Department of Mathematics and Statistics Concordia University, Montreal, QC, H4B 1R6

Email: barr@barrs.org, jkennison@clarku.edu, raphael@alcor.concordia.ca

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