Tychonoff spaces and a ring theoretic order on C(X)
aa r X i v : . [ m a t h . GN ] M a y Tychonoff spaces and a ring theoretic order on C ( X ) W.D. Burgess and R. Raphael
Abstract.
The reduced ring order ( rr -order) is a natural partial order on areduced ring R given by r ≤ rr s if r = rs . It can be studied algebraically ortopologically in rings of the form C( X ). The focus here is on those reducedrings in which each pair of elements has an infimum in the rr -order, and whatthis implies for X . A space X is called rr -good if C( X ) has this property.Surprisingly both locally connected and basically disconnected spaces sharethis property. The rr -good property is studied under various topological con-ditions including its behaviour under Cartesian products. The product of two rr -good spaces can fail to be rr -good (e.g., β R × β R ), however, the product ofa P -space and an rr -good weakly Lindel¨of space is always rr -good. P -spaces, F -spaces and U -spaces play a role, as do Glicksberg’s theorem and work byComfort, Hindman and Negrepontis. Introduction.
In a reduced ring, a ring with no non-zero nilpotent elements, such as C( X ),there is a partial order that generalizes the natural partial order on a boolean ring.The order relation is defined as r ≤ rr s if r = rs . The study of this order, herecalled the rr -order for the reduced ring order , goes back at least to 1958 in [ S ].Since then it has been studied at various times (see [ Ch ] and [ B ], for example), butmost recently in [ BR1 ] and [
BR2 ]. In these papers some of the most interestingexamples and results are about rings of the form C( X ).It is rare for a pair of elements in a reduced ring R to have a supremum inthe rr -order and the most natural generalization of the boolean ring case is wherethe ring has, for every pair of elements r, s ∈ R , an infimum in the rr -order, noted r ∧ rr s , i.e., when R is a lower semi-lattice in the order. Such rings are called rr -good . A space X is called rr -good if the ring C( X ) is rr -good. The theme of thispaper is the study of spaces that are rr -good and those that are not.In the sequel, all topological spaces will be assumed to be Tychonoff spaces. Not all spaces are rr -good but those that are form a surprisingly diverse familythat includes locally connected spaces and those that are basically disconnected.To find a topological characterization of rr -good spaces would seem an unrealistictask but much can be said about them. There are connected spaces that are not Mathematics Subject Classification.
Key words and phrases. reduced ring order, locally connected, basically disconnected, semi-lattice.The authors thank the referee for helpful observations as well as for suggestions about thepresentation.DOI 10.1016/j.topol.2020.107250. rr -good (see Theorem 2.9, below) and even connected, compact metric spaces thatare not rr -good ([ BR1 , Theorem 3.5(2)]).The paper is divided into five sections. The first gives some basic results andexamples. The second deals with when a product of two spaces is rr -good. If X × Y is rr -good it is easy to see that X and Y are also rr -good. The converse, false ingeneral, turns out to be a rich subject.If the real line R is partitioned into two complementary dense subspaces, neithercan be rr -good. The third section shows that R is quite different. Complementarydense subspaces of the plane are found, one of which is rr -good.Section 4 examines basically disconnected and U -spaces with an emphasis onseparation properties and their consequences for C( X ).Section 5 looks at a sufficient condition, called the B-property (for boundaryproperty), for a space to be rr -good. Basically disconnected spaces that are notdiscrete do not have the B-property. Here it is shown how to find connected rr -goodspaces without the B-property.
1. The definition of rr-good spaces: basic properties and examples.
To recall: a ring C( X ) is partially ordered by the relation f ≤ rr g if f = f g .When h ≤ rr f and h ≤ rr g this is abbreviated to h ≤ rr f, g . The following facts areobvious but are basic tools underlying many of the results used below. Lemma . In a ring C ( X ) , (1) if f ≤ rr g then f and g coincide on cl (coz f ) ;(2) if h ≤ rr f, g then coz h ⊆ z( f − g ) ∩ coz f . It is clear that a free union of rr -good spaces is rr -good since a product of rr -good rings is rr -good. The following proposition quotes results that show how rr -good spaces can be found from a given rr -good space. Proposition . (1) ( [ BR1 , Proposition 3.10] ) A cozero set in an rr -goodspace is rr -good.(2) ( [ BR1 , Proposition 3.9] ) The ring C ( X ) is rr -good if and only if C ∗ ( X ) is rr -good, i.e., a space X is rr -good if and only if βX is rr -good. The two main classes of examples of rr -good spaces are summarized here. Examples . (1) [ BR1 , Theorem 3.5(1)] If X is a locally connected spacethen X is rr -good.(2) [ NR , before Ex. 3.2] If X is basically disconnected then X is rr -good. The second case will be expanded upon in Section 4.It is not true that a quotient space of an rr -good space has to be rr -good. Example . The space β N is rr -good but its quotient space N ∪ {∞} (theone-point compactification) is not. Proof.
The space N ∪ {∞} is not rr -good by [ BR1 , Proposition 3.6], or seeLemma 3.1, below. (cid:3)
Sometimes quotients behave well.
Proposition . (1) If X is a locally connected space, all its quotient spacesare rr -good. (2) In particular, if X is a locally connected pseudocompact space thenall its continuous images are rr -good. YCHONOFF SPACES AND A RING THEORETIC ORDER ON C( X ) 3 Proof. (1) All the quotients spaces of a locally connected space are locallyconnected. (2) This is by [ W , page 223]. (cid:3) The long line is an example of part (2) of the proposition.Note also that every space, rr -good or not, can be embedded in a direct productof copies of a closed interval, a compact, locally connected ( rr -good) space.This section closes with a pair of illustrative examples.The space Λ = β R \ ( β N \ N ) of [ GJ , 6P] is pseudocompact and rr -goodbecause β Λ = β R is rr -good, but it is known not to be locally connected [ W , pp221,222]. This space will appear again in Example 2.11 and at the end of Section 2.On the other hand the pseudocompact Tychonoff plank T is not rr -good. If itwere, β T would also be rr -good. However, β T has a clopen subset which is home-omorphic to the one-point compactification of N , a space which is not rr -good,showing β T is not rr-good by Proposition 1.2(1).
2. Product spaces and the rr-order.
In this section the question of rr -good product spaces will be examined. It willbe easy to see that if a product is rr -good, so are its factors. The converse, false ingeneral, will take up much of the section. Proposition . Suppose Y is a retract of an rr -good space X . Then, Y is rr -good. Proof.
Let φ and ψ be continuous functions X φ → Y ψ → X with ψφ = X and f, g ∈ C( X ). It is easy to see that if h = f ψ ∧ rr gψ then hφ = f ∧ rr g . (cid:3) Corollary . If X and Y are spaces such that X × Y is rr -good, then X and Y are rr -good. As already mentioned, the converse is false but there are some cases wherethere are positive results.
Proposition . (1) If { X α } α ∈ A are locally connected spaces all but finitelymany of which are connected then Q α ∈ A X α is rr -good.(2) If { X , . . . , X n } is a finite set of P -spaces then Q ni =1 X i is rr -good. Proof. (1) These products are locally connected and, hence, rr -good. (2) Afinite product of P -space is a P -spaces and, hence, rr -good. (cid:3) As examples, all euclidean spaces are rr -good. Other types of rr -good productswill be found at the end of this section.The following will show that if a space X has enough clopen sets and is rr -good,then it is basically disconnected. This will play a role later in this section and againin Section 4. Proposition . Let X be a space which has a clopen π -base. If X is rr -goodthen X is basically disconnected. Proof.
For f ∈ C( X ) it will be shown that cl (coz f ) is clopen. Since X is rr -good, h = ∧ rr ( − f ) exists. Because h ≤ rr , h is an idempotent and coz h = D is clopen. Moreover, h = h = h ( − f ) implies that hf = . When E ⊆ z( f ) isclopen, let the idempotent e have cozero set E . It follows that e ≤ rr , ( − f ) and,from this, e ≤ rr h , giving E ⊆ D . Hence, D is the unique largest clopen set in z( f ). W.D. BURGESS AND R. RAPHAEL
If cl (coz f ) = X \ D then, because of the clopen π -base, there would be anon-empty clopen set in ( X \ D ) \ cl (coz f ). This would contradict the fact that D is the maximal clopen set in z( f ). (cid:3) A first step in finding examples is to recall two results of Negrepontis.
Proposition . (1) [ N , Theorem 7.3] For any P -space X there exists anextremally disconnected space Y for which X × Y is not an F -space. (2) [ N , The-orem 6.3] The product of a P -space and a compact basically disconnected space isbasically disconnected. Corollary . (1) If X is a P -space and Y is extremally disconnected suchthat X × Y is not an F -space, then X × Y is not rr -good. (2) If X is a P -spaceand Y is compact and basically disconnected, then X × Y is rr -good. Proof. (1) If X × Y were rr -good, Proposition 2.4 would say that it is basicallydisconnected and, hence, an F -space. (2) Proposition 2.5 (2) gives the result. (cid:3) The case where neither space is a P -space can also be dealt with as follows. Theorem . Let X and Y be spaces such that each has a clopen π -base andare not P -spaces. The space X × Y is not rr -good. Proof.
Every non-empty open set in X × Y contains a non-empty clopen. If X × Y were rr -good it would be basically disconnected by Proposition 2.1, hencean F -space, so one of X and Y would be a P -space by [ Cu , Theorem p. 51] or by[ GJ , 14Q.1] (cid:3) Theorem 2.7 yields families of examples.
Examples . If X and Y are basically disconnected but not P -spaces, then X and Y are rr -good but X × Y is not rr -good. As an illustration, β N × β N isnot rr -good. Another example of a product of rr -good spaces that is not rr -good is foundin the next result. It is of a quite different sort than in Examples 2.8, indeed, thefactors are connected. The functions needed in the proof are best presented by adescription of their graphs. Theorem . The space β R × β R is not rr -good. Proof.
Consider a band of width 2 centred on the diagonal D = { ( x, x ) | x ∈ R } in R × R , bounded by two lines parallel to D , L above and L below.Functions f, g ∈ C( R × R ) will be defined.(1) In the region above and including line L , f ( x, y ) = 3 and g ( x, y ) = 2.(2) In the region below and including L , f ( x, y ) = g ( x, y ) = 0.(3) Let L be the line parallel to D , midway between D and L . On any line M perpendicular to D , let f go linearly from 3 to 0 as ( x, y ) goes from L to L . Similarly, g will go linearly from 2 to 0 on M .(4) Everywhere below L both f and g will be 0 except where indicated below.(5) For each n ∈ N consider a disk ∆ n of radius 1/4 around ( n, n ). Thefunctions f and g will coincide on ∆ n and their graphs there will be aregular cone of height 1 and centre ( n, n ). YCHONOFF SPACES AND A RING THEORETIC ORDER ON C( X ) 5 There are several claims to be proved.
Claim 1:
Both f and g extend to β R × β R .As is customary in R × R , the first factor is the horizontal axis and the secondthe vertical one.It must be shown that the oscillation condition of [ W , Theorem, page 200] issatisfied so that f and g can be extended to β R × β R .It is readily seen that the functions f and g are uniformly continuous becauseof the repeated patterns along the diagonal. This means that for every ε > δ > | f ( x, y ) − f ( u, v ) | < ε if k ( x, y ) , ( u, v ) k < δ . Now fix ( x , y ), set ζ = (1 / √ δ and consider the vertical lines through x + mζ and horizontal linesthrough y + nζ , with m, n ∈ Z . Set U = S m ∈ Z ( x + mζ, x + ( m + 1) ζ ), a unionof intervals along the horizontal axis; and V = S n ∈ Z ( y + nζ, y + ( n + 1) ζ ), aunion of intervals along the vertical axis.Similarly, construct U and V using { x + (1 / m ) ζ } and { y + (1 / n ) ζ } , m, n ∈ Z . Then the open sets U × V and U × V cover R × R and make a gridsatisfying the oscillation conditions. Since this can be done for all ε > f and g can be extended to elements of C( β R × β R ), say F and G , respectively. Claim 2: If H = F ∧ rr G then for n, m ∈ N , n = m, H ( n, m ) = 0. In the casewhere n > m , this holds because both f and g vanish at ( n, m ). In the case where n < m , this holds because f and g do not agree at ( n, m ) (Lemma 1.1(2)). Claim 3:
For n ∈ N let h n ∈ C( R × R ) be the function whose graph is thecone defined for ( n, n ). The function h n extends to β R × β R because of the samegrid as used for f and g . Let H n denote its extension to β R × β R . Observe that H n ≤ rr F, G since h n ≤ rr f, g on the dense subset R × R . Claim 4:
It follows from Claim 3 that H ( n, n ) = 1 for all n ∈ N since H n ( n, n ) = 1and H n ≤ rr H . This means that H , restricted to N × N , is the Kronecker deltafunction, which is shown in [ W , p. 196] not to extend to β N × β N . This is acontradiction. (cid:3) Corollary . For any m, n ≥ , β ( R m ) × β ( R n ) is not rr -good. Proof.
The space R is a retract of R m and, hence, β R is a retract of β ( R m )and, similarly, β R is retract of β ( R n ). From this, β R × β R is a retract of β ( R m ) × β ( R n ). The result follows from Theorem 2.9 and Proposition 2.1. (cid:3) It would be interesting to know if R × β R is rr -good or not. The methods usedabove do not apply to this space.For spaces X and Y it is possible for β ( X × Y ) to be homeomorphic to βX × βY ,where the homeomorphism does not fix X × Y (see [ W , 8.18] for such an example).In the case of a homeomorphism, all the spaces X × Y , β ( X × Y ) and βX × βY are simultaneously rr -good or none of them is. The latter possibility is illustratedin the next example. Example . There is a connected non-compact rr -good pseudocompact spacewhose product with itself is pseudocompact but not rr -good. Proof.
The example is the space Λ = β R \ ( β N \ N ) mentioned at the end ofSection 1. It is rr -good but β Λ × β Λ = β R × β R is not rr -good. On the other hand,Λ × Λ is pseudocompact by [ W , Proposition, p. 203]. Hence, Glicksberg’s theoremapplies showing that β (Λ × Λ) = β Λ × β Λ. This is not rr -good and therefore Λ × Λis not rr -good either. (cid:3) W.D. BURGESS AND R. RAPHAEL
This section ends with some rr -good products. Unlike previous examples, theones to be presented here need not be locally connected or basically disconnected. Asimple lemma, whose proof follows by direct point-wise calculations, will be useful. Lemma . Let Y be an rr -good space and X any space. Suppose f, g ∈ C ( X × Y ) . For each x ∈ X , let f x , g x ∈ C ( Y ) be given by f x ( y ) = f ( x, y ) and g x ( y ) = g ( x, y ) . Set h x = f x ∧ rr g x and let h be defined by h ( x, y ) = h x ( y ) , for all x, y . If h is continuous on X × Y then h = f ∧ rr g . Recall that a space X is weakly Lindel¨of if for any open cover { U α } α ∈ A , thereis a countable subfamily { U α n } n ∈ N with S n ∈ N U α n dense in X . In the following,a result of Comfort, Hindman and Negrepontis ([ CHN ]) will be crucial.
Theorem . Let X be an arbitrary P -space and Y an rr -good weakly Lin-del¨of space. The space X × Y is rr -good. Proof.
Fix f, g ∈ C( X × Y ). By [ CHN , Lemma 3.2], each point in X × Y lies in an open set of the form U × V , U open in X , V open in Y , such that for x, x ′ ∈ U and all y ∈ V , f x ( y ) = f x ′ ( y ) (the notation is as in the lemma). Thereare such open sets for g as well and, by taking intersections, it may be assumedthat these open sets work for both functions. Moreover, since X is a P -space, itmay also be assumed that U is clopen. An open set U × V where U is clopen andthe [ CHN ] properties hold for both f and g will be here called a tile .Fix p ∈ X . For each y ∈ Y there is a tile U × V with ( p, y ) ∈ U × V . Hence, theset of tiles { U β × V β } β ∈ B , with p ∈ U β , is such that S β ∈ B V β = Y . By the weaklyLindel¨of property, there is a countable subset { V β n } n ∈ N whose union, V p , is densein Y . Put A p = T n ∈ N U β n . Since X is a P -space, A p is clopen. For all x ∈ A p , f x = f p are equal on the dense open set V p . Hence, f x = f p on Y . Similarly g x = g p on Y . Put h p = f p ∧ rr g p and notice that h p = f x ∧ rr g x , for all x ∈ A p .Now consider p, p ′ ∈ X . If A p ∩ A p ′ = ∅ , then h p = h p ′ . Indeed, for x ∈ A p ∩ A p ′ , f x = f p = f p ′ and g x = g p = g p ′ . Define h ( x, y ) = h p ( y ) whenever x ∈ A p . This iswell-defined and continuous on all the elements of the open cover of { A p × Y } p ∈ X .Hence, h = f ∧ rr g by Lemma 2.12. (cid:3) There are many examples of rr -good Lindel¨of spaces Y which can be used inProposition 2.13, for example R . For any non-discrete P -space X , X × R is rr -goodbut neither locally connected nor basically disconnected. An example where the rr -good space Y is weakly Lindel¨of but not Lindel¨of is Λ, described at the end ofSection 1. Another is found in [ LR , Example 2, p. 237].
3. A partition of R into two dense subspaces, one rr-good: this isimpossible in R. The first thing to note is that if the real line R = A ∪ B , with A ∩ B = ∅ and A and B both dense, then neither A nor B is rr -good. This is a consequence of thefollowing. Lemma . [ BR1 , Proposition 3.6]
Suppose, in a space X , there is a sequence { D n } n ∈ N of pairwise disjoint clopen sets such that U = S n ∈ N D n is not closed andthere is x ∈ Fr U (the boundary or frontier) such that every neighbourhood of x meets all but finitely many of the D n . Then, X is not rr -good. To use the lemma in the case of A and B in R , it suffices to take a convergentincreasing sequence { a n } n ∈ N in, say, A and intersperse it with a sequence from B . YCHONOFF SPACES AND A RING THEORETIC ORDER ON C( X ) 7 Subsets of R will now be constructed to show a quite different situation in theplane. Definition . A line y = mx + b in R is called matched if m, b ∈ Q and m = 0. The graph of such a line is denoted L m,b . Lemma . Consider a matched line L m,b in R given by y = mx + b , where m = 0 and m, b ∈ Q . Then if ( p, q ) ∈ L m,b , both p, q ∈ Q or both are irrational. Proof. If x ∈ Q then y = mx + b ∈ Q . If x / ∈ Q then y = mx + b ∈ Q wouldimply mx ∈ Q , but m ∈ Q and x / ∈ Q , which is impossible. (cid:3) It is also useful to note that if ( a, b ) and ( c, d ) are such that a, b, c, d ∈ Q , a = c ,then the line joining these points is a matched line. Theorem . Consider the following two subsets of R : B = [ m,b ∈ Q ,m =0 L m,b and A = R \ B .
Then,(1) B is dense in R , locally connected and, hence, rr -good.(2) A is dense in R and has a basis of clopen sets. It is not rr -good. Proof. (1) Since any open set in R contains points where both coordinatesare rational, B is dense in R . Notice that B also contains points ( a, b ) where both a and b are irrational, but not all such points.Consider a point ( a, b ) in B and an open disk C with centre ( a, b ). Supposethat U and V are open sets of R such that U ∪ V ⊇ C ∩ B , U ∩ V ∩ C ∩ B = ∅ , U ∩ C ∩ B = ∅ and V ∩ C ∩ B = ∅ . In other words assume that there is a partitionof C ∩ B . Choose points ( p, q ) ∈ U ∩ C ∩ B and ( u, v ) ∈ V ∩ C ∩ B , p, q, u, v ∈ Q , u = p . The line segment joining these two points will lie in C ∩ B but this linesegment is connected in R , which is impossible. Hence, B is locally connectedand, hence, rr -good by [ BR1 , Theorem 3.5(1)]. (It can be seen that B is evenarcwise connected.)(2) The set A contains all points ( a, b ) where one coordinate is rational andthe other irrational, as well as some points where both coordinates are irrational.This shows that A is dense in R . Moreover, for any ( a, b ) ∈ A and any open disk C with centre ( a, b ) there is a quadrilateral inside C containing ( a, b ) bounded bymatched lines. The interior of such a quadrilateral, intersected with A , is a clopenset in A .Since A has a basis of clopen sets and has convergent sequences, it is not rr -good by Lemma 3.1. (cid:3) There are similar constructions in R n , n >
4. Some separation properties and C ( X ) . Two sorts of reduced rings will make an appearance in this section. The def-initions are recalled here and, in the case of C( X ), the corresponding topologicalnotions will follow. Definition . (1) A ring R is called weakly Baer or wB if, for each r ∈ R ,ann r is generated by an idempotent e = e . (2) A ring R is called almost weaklyBaer or awB if, for each r ∈ R , ann r is generated by a set of idempotents. W.D. BURGESS AND R. RAPHAEL
In the literature the names “pp-ring” and “almost pp-ring” are also used forwB and awB rings, respectively.The first thing to note is the following.
Lemma . [ BR1 , Theorem 2.6]
An awB ring is rr -good if and only if it iswB. Not all awB rings are wB.
Example . [ NR , Example 3.2] The ring C ( β N \ N ) is awB but not wB. Even though awB rings need not be rr -good, a topological description of themnicely parallels that for wB rings of the form C( X ), and is given here.The equivalence of the first two statements in the following is mentioned in[ NR ] but is also proved here. Proposition . The following three statements about a space X are equiv-alent. (1) X is basically disconnected; (2) C ( X ) is a wB ring; and (3) if U is acozero set and V an open set with U ∩ V = ∅ then U and V can be separated by aclopen set. Proof. (1) ⇒ (2): Consider f ∈ C( X ) and let D = X \ (cl (coz f )), a clopenset, and e = e ∈ C( X ) such that coz e = D . For any g ∈ ann f , ge = g and f e = 0. Hence, ann f = e C( X ). (2) ⇒ (3): Let U = coz f and V be open with U ∩ V = ∅ . Since ann f = e C( X ) for some e = e , the clopen set D = coz e is suchthat coz f = U ⊆ X \ D . For every g ∈ C( X ) with coz g ⊆ V , f g = implying thatcoz g ⊆ D . Thus, the clopen set D separates U and V . (3) ⇒ (1): If U = coz f ,put V = int ( X \ U ). There is a clopen set D with coz f ⊆ D and V ⊆ X \ D . Itfollows that cl U = D . (cid:3) The equivalence of (1) and (2) in the next result was obtained in [ AE , The-orem 2.4], but the proof here is more direct. U -spaces were introduced in [ GH ];they are spaces X such that, for each f ∈ C( X ), there is a unit u ∈ C( X ) with f = | f | u . Proposition . The following statements for a space X are equivalent.(1) X is a U -space; (2) C ( X ) is an awB ring; and (3) if U and V are cozerosets with U ∩ V = ∅ then U and V can be separated by a clopen set. Proof. (1) ⇒ (2): Let = f, g ∈ C( X ) with f g = . Replace f by k = −| f | and g by l = | g | ; the cozero sets do not change. There is a unit u such that k + l = | k + l | u = ( − k + l ) u . Hence, for x ∈ coz k , u ( x ) = − x ∈ coz l , u ( x ) = 1. Since u is a unit, there is a clopen set D such that for x ∈ D , u ( x ) > x / ∈ D , u ( x ) <
0. From this, coz k = coz f ⊆ X \ D and coz l = coz g ⊆ D .Put e = e with coz e = D . Then, f e = and g = eg , showing that ann f isgenerated by idempotents.(2) ⇒ (3): Let U = coz f and V = coz g be such that U ∩ V = ∅ . The product f g = . Since C( X ) is awB there are e i = e i and l i ∈ C( X ), i = 1 , . . . , k , witheach e i such that f e i = and g = P ki =1 e i l i . Since D = S ki =1 coz e i is clopen, thereis e = e with coz e = D . From this, f e = and g = ge . The clopen set coz e separates U and V .(3) ⇒ (1): It must be shown that for any f ∈ C( X ) there is a unit u with f = | f | u . If f does not change sign in coz f , the unit can be ± . Otherwise, let YCHONOFF SPACES AND A RING THEORETIC ORDER ON C( X ) 9 f + be defined by f + ( x ) = f ( x ) if x ∈ coz f and f ( x ) > f + ( x ) = 0 for other x . Similarly, f − is defined. Since coz f + ∩ coz f − = ∅ , there is a clopen set D with coz f + ⊆ D and coz f − ⊆ X \ D . Let e = e be such that coz e = D and u = e − ( − e ), a unit. From this, f = f + + f − = f + u − f − u = | f | u . (cid:3)
5. A sufficient but not a necessary condition for rr-good.
We begin by recalling the definition of the B-property from [
BR1 , Definition3.3]. It is a sufficient condition for a space to be rr -good ([ BR1 , Corollary 3.4]).It is implied by local connectedness. However, it is known not to be a necessarycondition; a topic expanded upon here.
Definition . In a space X let { U α } α ∈ A be any family of non-empty cozerosets in X with the following property: for α = β in A , ( Fr U α ) ∩ U β = ∅ . The space X is said to satisfy the B-property (for boundary property ) if the following holds foreach such family of cozero sets. Let z ∈ Fr ( S α ∈ A U α ). For every neighbourhood N of z there is β ∈ A such that N ∩ Fr U β = ∅ .The motivation for this definition is as follows: Suppose in C( X ) that, for f, g ∈ C( X ), there are non-zero rr -lower bounds { h α } α ∈ A for f and g . Then, byLemma 1.1, the set { coz h α } α ∈ A satisfies the demands of Definition 5.1.The purpose here is to find connected rr -good spaces without the B-property.Before doing that, the next proposition shows that, at the other extreme, it is easyto find basically disconnected spaces without the B-property. Proposition . If X is a space that has the B-property then each union ofclopen sets is clopen. If, in addition, X has a clopen π -base, it is discrete. Proof.
Any set { U α } α ∈ A of clopen sets satisfies the conditions of Defini-tion 5.1. Set U = S α ∈ A U α . If x ∈ Fr U , any neighbourhood of x would meet Fr U α , for some α . However, Fr U α = ∅ and, hence, Fr U = ∅ . For the second part,any open V in X has a union of clopen sets dense in it. From the first part, V isclopen. (cid:3) Any basically disconnected space X which is not discrete is rr -good and doesnot have the B-property.The next proposition is the key tool for the construction of connected examples. Proposition . Let X be a space which is not compact. Let { U α } α ∈ A be aninfinite family of pairwise disjoint non-empty cozero sets of X . For α ∈ A , let f α be such that f α ∈ C ( X ) such that coz f α = U α , for all x ∈ X , ≤ f α ( x ) ≤ andfor some k α ∈ U α , f α ( k α ) = 1 . Assume that these data also satisfy the followingproperties:(i) for α = β , ( Fr X U β ) ∩ U α = ∅ ,(ii) K α = cl X U α is compact for all α ∈ A ,(iii) the function f defined by f ( x ) = f α ( x ) for x ∈ U α and f ( x ) = 0 if x / ∈ U = S A U α is continuous on X ,(iv) K = f − ( { } ) is not compact in X .Then, βX does not have the B-property. Proof.
The condition (ii) says that K α is compact and so it and X ∗ = βX \ X are completely separated. There is u α ∈ C( βX ) such that u α | K α is constantly 1and u α | X ∗ is constantly 0. Now f α can be extended to βf α ∈ C( βX ). The product u α · βf α coincides with f α on U α and is 0 elsewhere. This shows that U α is a cozeroset in βX . Moreover, Fr X U α = Fr βX U α because K α is compact and, hence, alsoclosed in βX .It follows that { U α } A is a family of cozero sets in βX which satisfies the con-dition to test for the B-property.Since K is closed and not compact in X , P = (cl βX K ) ∩ X ∗ = ∅ . Now, extend f to βf .It follows that (coz βf ) ∩ X = U but also P ⊆ coz βf , since for p ∈ P , βf ( p ) = 1.Notice that any p ∈ P is in Fr βX U since any neighbourhood N of p with N ⊆ coz βf will meet X and thus N ∩ X ⊆ U . However, any such N will not meetany Fr X U α = Fr βX U α , contradicting the B-property. (cid:3) Corollary . Suppose that X satisfies the conditions of Proposition 5.3and that X is rr -good. Then, βX is rr -good and does not have the B-property.Moreover, βX is not locally connected. Proof.
Since X is rr -good, so is βX . Then Proposition 5.3 says that βX does not have the B-property. If βX were locally connected, it would have theB-property. (cid:3) Example . The connected space β R is rr -good and does not have the B-property. Proof.
The space R is rr -good. The cozero sets needed in the proposition canbe taken to be the intervals { ( n, n + 1) } n ∈ Z and, hence, Corollary 5.4 applies. (cid:3) Similarly, for any euclidean space R n , β R n is a connected rr -good space whichdoes not have the B-property. References [AE] H. Al-Ezeh,
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