#### Abstract

Working within a complete (not necessarily atomic) Boolean algebra, we use a sublattice to define a topology on that algebra. Our operators generalize *complement* on a lattice which in turn abstracts the set theoretic operator. Less restricted than those of Banaschewski and Samuel, the operators exhibit some surprising behaviors. We consider properties of such lattices and their interrelations. Many of these properties are abstractions and generalizations of topological spaces. The approach is similar to that of Bachman and Cohen. It is in the spirit of Alexandroff, Frolík, and Nöbeling, although the setting is more general. Proceeding in this manner, we can handle diverse topological theorems systematically before specializing to get as corollaries as the
topological results of Alexandroff, Alo and Shapiro, Dykes, Frolík, and Ramsay.

#### 1. Introduction

We begin with , a sublattice of a complete (not necessarily atomic) Boolean algebra . If is closed under arbitrary meets, it abstracts the closed sets of a topological space. If not, we introduce a Kurotowski closure operator to define the associated topological lattice. The operators we define generalize *complement* on a lattice which in turn abstracts the set theoretic operator. Less restricted than those of Banaschewski [1] and Samuel [2], the operators exhibit some surprising behaviors. We consider certain properties of such lattices and the implications for the properties of one lattice from those of another, when one is a sublattice of the other. Many of these properties are abstractions and generalizations of topological spaces.

The approach is similar to that of Bachman and Cohen [3, 4]. It is in the spirit of Alexandroff [5], Frolík [6], and Nöbeling [7], although the setting is more general. We generalize a variety of filter arguments used in paved space [8, 9] and in the theory of realcompactness [10, 11]. Proceeding in this manner, we can handle diverse topological theorems systematically before specializing to get as corollaries as the topological results of [5, 8, 12–14].

Section 2 provides some background material and generates a topology on an algebra by means of a sublattice. Section 3 defines operators and topological type properties for a lattice. Section 4 examines filter and measure behavior with respect to the operators. Section 5 looks at covering properties. Section 6 investigates the relationships between two lattices.

#### 2. Background, Terminology, and Notation

We work within a complete Boolean algebra with minimal element and maximal element . The usual operators are denoted by , and . is not necessarily atomic; equivalently, is not necessarily completely distributive [15].

(i) denote finitely additive zero-one measures on . (ii) and denote sublattices of containing and .(iii) denotes the algebra generated by .(iv) is the power set of the set . (v)The indices and index countable (finite or countably infinite) collections, while and index arbitrary ones.*Definition 2.1. *(i) is an **-filter** if and only if for all :(a),(b) and .

When there is no ambiguity, we simply say that is a **filter**.

(ii) An -filter is a **prime**-filter if and only if for all either or

(iii) An -filter is an **-ultrafilter** (or **ultra**) if and only if is a maximal -filter.

(iv) A filter is **fixed** if and only if . Otherwise is **free**.

(v) A filter has **cmp** (**countable meet property** or **countable intersection property**) if and only if for any *countable* subset of , .

*Remark 2.2. * It follows that (i) if and only if ,(ii)every ultrafilter is prime. In this paper, is not in any filter.

*Definition 2.3. *A measure on an algebra containing is **-regular** if and only if for all

Lemma 2.4. *There exist one-to-one correspondences between *(i)*zero-one measures on and prime -filters, *(ii)*-regular zero-one measures on and -ultrafilters [3, 4].*

Thus, analogous measure theoretic results may easily be derived from our filter statements.

We now topologize by means of a sublattice . The lattice elements themselves may not be sufficient to be used as open or closed sets. However, we will generate a topology.

*Definition 2.5. *Let be an algebra, a sublattice of and (i)(ii)(iii) is **closed** if and only if . (iv) denotes the set of closed elements of .

*Remark 2.6. *(i) is a Kurotowski closure operator.

(ii) is an interior operator.

(iii)

(iv) .

(v) is a lattice which is closed under arbitrary meets.

We observe that is an obvious abstraction of the closed sets in a topological space.

*Example 2.7. *Let be the lattice of zero sets in a completely regular topological space. Then is the lattice of closed sets [11].

*Definition 2.8. *Let be a meet semilattice (i.e., a subset of closed under finite meets). Consider the generated lattice . All terminology remains the same except that is a **prime ****-filter** if and only if (for ) implies that one of the .

Lemma 2.9. *
(i) There exist one-to-one correspondences between the prime, ultra-, fixed, and free filters on and those on ().**
(ii)-filters with cmp correspond with -filters with cmp. *

Thus we lose no generality in “treating” like a lattice.

*Example 2.10. *Let . is a meet semi-lattice since implies that . “An open subset in a topological space is **regularly open** if and only if is the interior of its closure” [16]. Thus the regularly open sets in a topological space form a meet semi-lattice.

Table 1 summarizes the notation used in this paper.

#### 3. Lattice Operators and Properties

In this section we define certain operators and lattice properties. These properties reduce to the conventional topological ones when the operator is taken to be *complement*.

*Definition 3.1. *Let be a meet semi-lattice containing and . We define to be a one-to-one operator on such that and . We define = and .

Proposition 3.2. *
(a). **
(b).**
(c). **
(d). *

*Proof. * (a)

(b) For all for all .

The proof of (c) and (d) follows readily.

From now on, we assume that is defined on a lattice . Then is a meet semi-lattice. By Lemma 2.9, we may “treat” like a lattice.

Corollary 3.3. *When is defined on , one has the following. *(a)*. *(b)

*Example 3.4. *Let and (the power set of ) with set union and intersection as the join and meet operations. Let with and as in Figure 1. The operator is defined by , and . We have and . In addition and Note that, unlike in [2], is not the maximal element disjoint from , and although is complemented (that is,

We now define various properties for . They generalize some of the definitions in point set topology, reducing to the conventional properties when is the power set of a set and is complement.

*Definition 3.5. *(i) is **compact** if and only if every -filter is fixed.

(ii) is **-compact** if and only if every -filter has cmp.

(iii) is an **I-lattice (P-lattice)** if and only if every (prime) -filter with cmp is contained in an ultrafilter with cmp.

(iv) is an **R-lattice** if and only if every filter which contains a fixed prime filter is also fixed.

(v) A topological space is an **I-space** if and only if the lattice of closed sets is an **I-lattice** [17].

Proposition 3.6. *As an immediate consequence of Definition 3.5, one has the following.*(i)* is compact is -compact is an I-lattice is a -lattice.*(ii)*The following are equivalent: (a) is compact (-compact),(b)every prime filter is fixed (has cmp),(c)every ultrafilter is fixed (has cmp). *

*Definition 3.7. *(i) is **-paracompact** if and only if whenever there exists there exists such that and .

(ii) When , we say that is **perfect**.

Proposition 3.8. *Every perfect lattice is -paracompact.*

*Proof. *Assume is perfect and . For each there exists such that . Let . Then and .

*Example 3.9. *The zero sets in a topological space are perfect (i.e., *complement generated* in the sense of [11]) and thus -paracompact.

*Definition 3.10. *Let . Then the following are given. (i) is if and only if for all there exists such that but .(ii) is **Hausdorff** if and only if for all with there exist such that (iii) is **regular** if and only if for all there exist such that , and

The following proposition provides an example.

Proposition 3.11. *Let be the power set of a topological space , let be the lattice of closed sets in and let be complement. is a (Hausdorff, regular) space if and only if is (Hausdorff, regular).*

*Proof. *Let be the lattice of closed sets in . Suppose is a lattice. Let be atoms in . Since , there exists such that but . By symmetry, there exists such that but . Thus is a space.

Now suppose is a topological space. Let . Then there exists an atom but . Thus for all atoms , and there exists such that but . Let Then , and thus is .

The proofs for Hausdorff and regular are similar.

Proposition 3.12. *Suppose and is regular. If is prime and , then .*

*Proof. *Suppose there exists such that . Let implies there exists But then and . By regularity, there exist such that and Now implies that . From **,** it follows that and thus . But implies that and thus is not prime.

*Example 3.13. *The closed sets in a regular topological space form an -lattice.

*Definition 3.14. * is **normal** if and only if for all there exist such that , and .

The following proposition demonstrates an example of an application.

Proposition 3.15. *Let be a topological space. Let be complement and let be the lattice of open sets. Then the following are equivalent.*(a)* is normal.*(b)* and .*(c)* (i.e., is extremally disconnected). *

*Proof. *Consider the following. (i) (a) implies (b): let . Then there exist such that (by normality). But and implies that .

(ii) (b) implies (c): let . Let so that . Then (by hypothesis), so that . Then , so that . Therefore by definition of .

(iii) (c) implies (b): let , so that . Since , then

(iv) (b) implies (a): are elements of .

Proposition 3.16. *Let . If is normal and is a prime -filter contained in two -ultrafilters and , then .*

*Proof. *Let and be two distinct ultrafilters and let . Then there exist with By normality, there exist such that But implies that , and implies that , so that , that is, . But implies that is not prime.

*Definition 3.17. * is **-normal** if and only if it is normal and -paracompact.

*Example 3.18. *A normal topological space is countably paracompact if and only if the lattice of closed sets is -paracompact [18]. Willard [16] calls such a space *binormal*.

Proposition 3.19. *Let be complement in . If is -normal and is a prime filter with cmp, then implies that has cmp.*

*Proof. *Let where is a prime -filter and is an -filter without cmp. Then there exists such that . Let and Thus there exist such that and . Now , so there exist two sequences such that for all . Since and is prime, we have or for all But implies that so for all And since implies that we have that does not have cmp.

*Remark 3.20. *If is a normal lattice that has the stronger property that whenever such that there exists such that and for all , then we need only to assume that

Corollary 3.21. *Let be complement in . If is -normal, then is a -lattice.*

#### 4. Behavior of Filters and Measures Under

In this section we look at the behavior of filters and measures with respect to . It is interesting to see an example where does not “behave as nicely” as complement.

*Definition 4.1. *Let . .

Proposition 4.2. *Let be a prime -filter. is a prime -filter.*

*Proof. *(a) since .

(b) and .

(c) Let , so that ; equivalently, . But then , and so . Thus (by (a), (b), and (c)), is a filter.

(d) Let . or or . Thus is prime.

*Definition 4.3. *A filter is **coultra** if and only if is -ultra. is **coregular** if and only if is coultra.

Proposition 4.4. * is -ultra if and only if is equivalent to for some (i.e., there exists such that ).*

*Remark 4.5. *Proposition 4.4 generalizes a theorem of [12] which we get by taking to be complement.

As demonstrated by the following example, we can associate measures with the prime filters on as usual, but they may lack some of the properties to which we are accustomed.

*Example 4.6. *Let . Let , with set union and intersection as the join and meet operations. Figure 2 defines the sublattice .

Define on as follows:

Incidentally, and for all . However does not necessarily = .

Now can be extended to all of by defining .

Let . Since is an -ultrafilter, is a coultra filter. is a prime filter so we have the associated measure where

Now consider that , so that But any element of whose measure is one, so is not -regular even though it is -coregular.

Note that

Thus, is not necessarily measure inverting (), preserving, increasing, or decreasing.

*Remark 4.7. *If and is measure inverting, then every -coregular measure is -regular. These concepts are equivalent when is complement.

From now on we assume that . Now is a lattice.

*Example 4.8. *See Examples 4.6 and 3.4.

Proposition 4.9. * is a prime -filter if is a prime -filter. (Please see Definition 4.1 and Proposition 4.2 for the converse.)*

*Proof. *(a) since .

(b) .

(c) .

(d) .

Corollary 4.10. *
(a) is a prime -filter if and only if is a prime -filter. **
(b) is a prime filter if is a coultra filter.*

#### 5. Covering Properties

In this section we define some covering properties for and show that they are analogous to the topological ones. In particular, when is taken to be complement, we get topological results as corollaries.

*Definition 5.1. * is **comax compact** if and only if every coultra filter is fixed. is **comax ****-compact** if and only if every coultra filter has cmp.

Proposition 5.2. *Every comax compact -lattice is compact.*

*Proof. *Let be a prime -filter. Form and extend it to , an -ultrafilter. is a prime -filter and fixed. implies that is fixed since is an -lattice. Thus is compact by Proposition 3.6.

Corollary 5.3. *If and is a comax compact regular lattice, then is compact.*

*Definition 5.4. * is **(prime, max, comax) complete** if and only if every (prime, ultra-, coultra) filter with cmp is fixed.

We have the implications in Figure 3.

Proposition 5.5. *If is (comax-)-compact, then it is (comax) complete if and only if it is (comax) compact.*

Proposition 5.6. *If is an R-lattice, then is prime complete if and only if it is comax complete.*

*Proof. *In the proof of Proposition 5.2, let have cmp.

Proposition 5.7. *If is a max complete I-lattice, then is complete.*

*Proof. *Let be a filter with cmp. Since is an I-lattice, can be extended to , an ultrafilter with cmp. is fixed because is max complete. Hence is fixed and is complete.

Corollary 5.8. *If L is a max complete P-lattice, then is prime complete.*

*Proof. *In the proof of Proposition 5.7, take to be a prime filter.

*Remark 5.9. *When is defined on (as when ) and , our definitions coincide with the conventional topological ones. (See Examples 3.4 and 4.6.) In particular may be taken to be complement.

Proposition 5.10. *Let be defined on with for all .*(a)* is compact if and only if . *(b)* is complete if and only if . *(c)* is -compact if and only if . *

*Proof. *We will prove only part (b). Parts (a) and (c) have similar proofs.

Suppose the condition holds. Let be a free -filter. Then implies that , so that there exists such that . But then and does not have cmp, so is complete.

Conversely, suppose that the condition does not hold. Then there exists such that but for any countable subset. Then and is a subbase for a filter . has cmp since implies that is free since implies that , and therefore, is not complete.

Corollary 5.11. *Let be a topological space. Take to be the closed sets and take to be complement. Then is compact (Lindelöf, countably compact) if and only if is compact (complete, -compact).*

Corollary 5.12. * is a realcompact I-space if and only if it is a Lindelöf space [14].*

Corollary 5.13. *If is regular, countably paracompact, and almost realcompact, then is realcompact [6].*

#### 6. Lattice Interrelations

In this section we investigate the implications between the properties of two lattices when one is a sublattice of the other.

Proposition 6.1. *Let *(a)*If is complete, then is complete. *(b)*If is prime complete, then is prime complete. *(c)*If is a P-lattice, then that is max complete implies that is prime complete.*

*Proof. * (a) Let be an -filter with cmp, and . There exists such that . As for all , we have for all . Now let , since is fixed. Thus is fixed and is complete.

(b) Let in (a) be prime. Then is prime.

(c) Since is a P-lattice, may be extended to an -ultrafilter with cmp. Since is max complete, is fixed, and hence and are fixed.

Corollary 6.2. *Let and let be a P-lattice. If is max complete, then is max and comax complete.*

Corollary 6.3. *Let and let be -normal. If is max complete, then is max and comax complete.*

Corollary 6.4. *If is a normal, countably paracompact space, then -replete implies -replete and realcompact implies -complete [13].*

The following proposition generalizes two results of Alexandroff [5].

Proposition 6.5. *Let . *(a)*If is compact, then is compact. *(b)*If is compact and normal, then is normal. *

*Proof. *(a) Let be an -filter. Let . The proof follows as in Proposition 6.1.

(b) Let , where , and of course for all . Now there must exist such that . (If not, would form a subbase for a free filter in a compact space.) By normality of , there exist such that and But then and and so is normal.

Proposition 6.6. *Let denote the smallest set containing and closed under countable meets. Let . If is -compact, then is -compact.*

Proposition 6.7. *Let If is complete, then is complete.*

*Proof. *Let be an -filter with cmp. Let has cmp since , and is complete so is fixed. Thus is fixed and is complete.

Corollary 6.8. *Let where is an I-lattice.*(a)*If is max complete, then is complete. *(b)*If is a comax complete R-lattice, then is complete.*

*Proof. *(a) Since is an I-lattice, that is max complete implies that is complete, which implies by Proposition 6.7 that is complete.

(b) Since is an R-lattice, its being comax complete implies that it is prime complete. Since is an I-lattice, is complete and thus is complete by Proposition 6.7.

*Definition 6.9. * is an -**P-lattice** if and only if every prime filter with cmp is contained in an ultrafilter with cmp.

*Definition 6.10. *(i) is **-normal** if and only if for all with there exist such that and

(ii) is **-paracompact** if and only if for each such that there exists a sequence such that and .

(iii) is **-normal** if and only if is -normal and -paracompact.

Proposition 6.11. *Let be complement. If is -normal and is a prime -filter with cmp contained in an -filter , then has cmp.*

*Proof. *Similar to the proof of Proposition 3.19.

Corollary 6.12. *Let be complement. *(a)*That is -normal implies that is an -P-lattice. *(b)*If is -paracompact and separates then is -paracompact.*

Corollary 6.13. *
(a) Let be an -P-lattice. That is max complete implies that is prime complete. **
(b) If in addition is an R-lattice, then that is comax complete implies that is prime complete.*

We get Frolík's [8] theorems as our final corollary.

Corollary 6.14. *
(a) If and is -normal and prime complete, then is prime complete. **
(b) Let be a normal space. If is almost realcompact and countably paracompact, then is realcompact.*

#### Acknowledgments

This paper is based on the Ph.D. thesis the author wrote under the supervision of the late George Bachman at The Polytechnic Institute of Brooklyn, now Polytechnic University of NYU. The author is grateful for Professor Bachman’s patient guidance and encouragement. This material was presented at The Second Annual Dr. George Bachman Memorial Conference at St. John's University Manhattan Campus on June 7, 2009. The author appreciates the comments and encouragement provided by Professors Keith Harrow, Pao-Sheng Hsu, and Noson Yanofsky and the questions raised by the anonymous reviewers. This paper is dedicated to the memory of Professor George Bachman.