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More on locales in which every open sublocale is z-embedded
Topology and its Applications 201 (2016) 110–123
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More on locales in which every open sublocale is z-embedded ✩ Themba Dube ∗ , Oghenetega Ighedo Department of Mathematical Sciences, University of South Africa, P.O. Box 392, 0003 Unisa, South Africa
a r t i c l e
i n f o
Article history: Received 31 December 2014 Accepted 8 June 2015 Available online 22 December 2015 MSC: 06D22 54C25 54C30 54D15
a b s t r a c t The locales which have the feature stated in the title (we shall call them Ozlocales) were introduced by Banaschewski and Gilmour as localic extensions of Blair’s Oz-spaces. In this note we give further characterisations of Oz-locales and their generalisations which we shall call weak Oz-locales. These characterisations include ring-theoretic ones which are new even for spaces. Among locales L which satisfy some weaker forms of normality, we characterise those for which βL is an Oz-locale. © 2015 Elsevier B.V. All rights reserved.
Keywords: Locale Oz-locale Sublocale Stone–Čech compactification Principal z-ideal
1. Introduction A Tychonoff space in which every open set is z-embedded is called an Oz-space. These spaces were introduced by Blair [8] who gave several characterisations of them. Subsequently they were also studied by other authors such as Lane [19] and Terada [27]; the latter concentrating mainly on characterising those Oz-spaces whose Stone–Čech compactifications are also Oz-spaces. Oz-spaces are also called perfectly κ-normal by some authors. On the other hand, Oz-locales (which generalise Oz-spaces in a natural and conservative way) were introduced by Banaschewski and Gilmour [7], and studied further in [4] where they are characterised in several ways. We shall here call them Oz-locales to be in line with such concepts as sublocale, Gδ -sublocale, and so on, which shall feature prominently in this discussion. A thorough literature review of Oz-spaces and ✩ This work is based on the research supported in part by the National Research Foundation of South Africa (Grant Number 93514). * Corresponding author. E-mail address: [email protected] (T. Dube).
http://dx.doi.org/10.1016/j.topol.2015.12.030 0166-8641/© 2015 Elsevier B.V. All rights reserved.
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Oz-locales shows that nowhere are these objects characterised in terms of their rings of continuous functions. One of our goals in this note is to give more characterisations of Oz-locales, including ring-theoretic ones which are new even for spaces. Among locales L which satisfy some weaker forms of normality, we characterise for which βL is an Oz-locale. These weaker forms of normality are localic versions of what Mack [20] calls weak δ-normality, almost normality introduced by Singal and Arya [25], and mild normality introduced by Singal and Singal [26]. Here is a brief outlay of the paper. We collect a few reminders in Section 2, concentrating mainly on those concepts which are not included in our main reference [24]. Following that, we present characterisations of Oz-locales and weak Oz-locales in Section 3, and in the last section we characterise when the Stone–Čech compactification is an Oz-locale. As mentioned above, we have put some normality condition on a locale L to obtain characterisations for βL to be an Oz-locale. We should remark though that, for an arbitrary L, if βL is an Oz-locale, then L does satisfy all the normality conditions we have imposed. In one of the corollaries we give necessary and sufficient conditions in terms of properties of λL (the regular Lindelöf reflection of L in the category Loc) for βL to be an Oz-locale. These of course do not have topological analogues. 2. Preliminaries 2.1. Frames Our reference for the general theory of frames and locales is [24]. Throughout the paper all frames are assumed to be completely regular. In certain instances we will mention complete regularity for emphasis. Let L be a frame. We view the Stone–Čech compactification of L as the frame of completely regular ideals of L. We denote by jL : βL → L the join map J → J. The right adjoint of jL is here denoted by rL . Recall that, for any a ∈ L, rL (a) = {x ∈ L | x ≺≺ a}. We write, as usual, Coz L for the set of cozero elements of L. If a, b ∈ Coz L, then rL (a ∨ b) = rL (a) ∨ rL (b); and if L is normal, then rL (x ∨ y) = rL (x) ∨ rL (y) for all x, y ∈ L. We shall frequently use the fact that Coz L is a normal σ-frame, as a consequence of which (see [6]) we have that whenever c ∨ d = 1 in Coz L, then there exist u, v ∈ Coz L such that u ≺≺ c, v ≺≺ d and u ∨ v = 1. A frame homomorphism h : L → M is coz-onto if for every d ∈ Coz M there is a c ∈ Coz L such that h(c) = d. By a point of L we mean a prime element, that is, an element p < 1 such that for any a and b in L, a ∧ b ≤ p implies a ≤ p or b ≤ p. We denote by Pt(L) the set of all points of L. We remark that, subject to appropriate choice principles (which we assume throughout), a compact regular frame has enough points, which means that every element is a meet of points. An element a ∈ L is called regular if a = a∗∗ . It is dense if a∗ = 0. A frame homomorphism is dense if the bottom element of its domain is the only element it maps to the bottom of its codomain. If h : M → L is dense onto, then (i) for any a ∈ M , h(a∗ ) = h(a)∗ , and (ii) for any b ∈ L, h∗ (b∗ ) = (h∗ (b))∗ . A consequence of this is that dense onto homomorphisms send regular elements to regular elements; as do their right adjoints. 2.2. The Lindelöf coreflection An ideal of Coz L is a σ-ideal if it is closed under countable joins. The regular Lindelöf coreflection of L (see [21] for details), denoted λL, is the frame of σ-ideals of Coz L. The join map λL : λL → L is a dense onto
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frame homomorphism, and is the coreflection map to L from completely regular Lindelöf frames. For any a ∈ L we write JaK = {c ∈ Coz L | c ≤ a}, and recall that the right adjoint of λL is given by (λL )∗ (a) = JaK. The map ηL : βL → λL
ηL (I) = I σ ,
given by
where · σ signifies σ-ideal generation in Coz L, is a dense onto frame homomorphism. In fact, ηL : βL → λL ηL
λ
L L is exactly realises the Stone–Čech compactification of λL. Furthermore, the composite βL −→ λL −→ the map jL : βL → L.
2.3. Sublocales Let L be a locale. As in [24], S(L) designates the lattice of all sublocales of L. We use the notation o(a) and c(a), respectively, for open and closed sublocales of L determined by a ∈ L. If a is regular we say o(a) is regular-open and c(a) is regular-closed. If a ∈ Coz L, we say o(a) is a cozero sublocale and c(a) is a zero-set sublocale; the latter term comes from [18]. Following [17], we say an element of L is δ-regular if it is the join of countably many elements each rather below it. Cozero elements are δ-regular. Following [17] again, but this time upside down, we say a sublocale S is a regular Gδ -sublocale if it is the intersection of countably many of its closed neighbourhoods. In [17] this is stated “dually” because the authors are working with the frame S(L)opp and not the co-frame S(L). We have thus used the hyphen to distinguish ∞ between the two terminologies. By a Gδ -coz-sublocale of L we mean a sublocale of the form o(cn ) where each cn ∈ Coz L. Of course a Gδ -sublocale is one of the form
∞
n=1
o(an ) where the an are arbitrary elements
n=1
in L. By a neighbourhood of a sublocale S ⊆ L we mean a sublocale T ⊆ L such that S ⊆ o(a) ⊆ T for some a ∈ L. That is, a sublocale whose interior contains S. A cozero neighbourhood of S is any o(a) ⊇ S with a ∈ Coz L. We say a collection G ⊆ S(L) is a neighbourhood base for S if each member of G is neighbourhood of S and every neighbourhood of S contains a member of G. The closure of a sublocale S will be denoted by S. The reader will recall that S
S=↑
and
o(a) = c(a∗ ).
The smallest sublocale of L is denoted by O. We say sublocales S and T are disjoint if S ∩ T = O. 2.4. Function rings Our approach to pointfree function rings is that of [3]. For any I ∈ βL, the sets M I = {α ∈ RL | rL (coz α) ⊆ I}
and
O I = {α ∈ RL | rL (coz α) ≺ I}
are ideals in the ring RL. For any a ∈ L we abbreviate M rL (a) as M a , and remark that M a = {α ∈ RL | coz α ≤ a}. Observe that if M a = M b for any a, b ∈ L, then a = b by complete regularity. The maximal ideals of RL are precisely the ideals M I , for I ∈ Pt(βL) (see [10]). The annihilator ideals in RL are precisely the ideals M a∗ , for a ∈ L [11, Lemma 3.1]. In particular, for any α ∈ RL, Ann(α) = M (coz α)∗ . Call an ideal of a
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ring A an element annihilator if it is of the form Ann(a) for some a ∈ A. Element annihilators in RL are exactly the ideals M c∗ , for c ∈ Coz L. 2.5. z-Ideals Let A be a commutative ring with identity, and let Max(A) denote the set of maximal ideals of A. We set M(a) = {M ∈ Max(A) | a ∈ M }
and
M (a) =
M(a).
Recall from [22] that an ideal I of A is called a z-ideal if, for every a, b ∈ A, a ∈ I and M(a) = M(b)
=⇒
b ∈ I.
Every maximal ideal is a z-ideal, and intersections of z-ideals are z-ideals. 3. Some characterisations of Oz-locales Let us start by recalling the pertinent definitions. First, an element s ∈ L is called coz-embedded if the frame homomorphism L → ↓s, given by x → s ∧ x, is coz-onto. The locale L is an Oz-locale if every a ∈ L is coz-embedded. We shall frequently use one or the other of the various characterisations, which we now record in the following proposition. The proofs are in [7] and [4]. For a locale L, the symbol L∗ denotes the sub-σ-frame of L generated by regular elements of L. Proposition 3.1. The following are equivalent for a completely regular locale L. (1) (2) (3) (4) (5) (6)
L is an Oz-locale. Every dense element of L is coz-embedded. For every a ∈ L, a∗ ∈ Coz L. For every regular a ∈ L, a ∨ a∗ ∈ Coz L. For all disjoint a, b ∈ L, there are disjoint c, d ∈ Coz L such that a ≤ c and b ≤ d. L∗ is a regular σ-frame.
As mentioned in the abstract, some of the characterisations we shall present are ring-theoretic. So we first introduce some terminology. It may seem odd that the rings we are about to define we will call Oz-rings, but it will soon be apparent why such a moniker is appropriate. Definition 3.2. An ideal of a ring A is a principal z-ideal if it is of the form M (a) for some a ∈ A. We say a ring is (i) an Oz-ring if every annihilator ideal of the ring is a principal z-ideal. (ii) a splitting ring if for any annihilator ideals I and J with I ∩ J = 0, there exist principal z-ideals U and V with U ∩ V = 0 such that I ⊆ U and J ⊆ V . A simple example of an Oz-ring is any integral domain with zero Jacobson radical, such as the ring Z. The following lemma gives a description of principal z-ideals in RL. Lemma 3.3. The principal z-ideals of RL are precisely the ideals M c , for c ∈ Coz L.
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Proof. It suffices to show that, for any α ∈ RL, M (α) = M coz α . It is shown in the proof of [12, Lemma 3.2] that, for any I ∈ βL, MI =
{M J | J ∈ Pt(βL) and I ⊆ J}.
Since for any J ∈ Pt(βL) we have α ∈ M J if and only if rL (coz α) ⊆ J, it follows that M coz α = M
rL (coz α)
{M J | J ∈ Pt(βL) and rL (coz α) ⊆ J} = {M J | J ∈ Pt(βL) and α ∈ M J }
=
= M (α), which proves the result. 2 Recall that two sublocales S and T of a locale L are said to be completely separated [2] if they have disjoint zero-set sublocale neighbourhoods. In particular, for any a, b ∈ L, the sublocales c(a) and c(b) are completely separated if and only if there exist c, d ∈ Coz L such that c ∧ d = 0 and a ∨ c = 1 = b ∨ d. Thus, if the closed sublocales c(a) and c(b) are completely separated, then there exists d ∈ Coz L such that d ≺ a and d ∨ b = 1. Indeed, if c and d are cozero elements with c ∧ d = 0 and a ∨ c = 1 = b ∨ d, then d ≺ a (with c as a witness for this) and d ∨ b = 1. A locale L is called mildly normal [17] if whenever a, b are regular elements such that a ∨ b = 1, then there exist u, v ∈ L such that u ∧ v = 0 and a ∨ u = 1 = b ∨ v. In a mildly normal locale every δ-regular element is a cozero element (see Remark 3.2(4) in [17]). In fact, this holds in what are called almost normal locales which include the mildly normal ones. The proof of one implication in the following result is modelled on that of (i) ⇒ (v) of [19, Theorem 1.1]. Also, we call the attention of the reader to the proof Banaschewski and Gilmour of [5, Lemma 1] which states (we are not quoting it verbatim) that If a and b are elements of a regular σ-frame, then there exist u and v such that u ∧ v = 0 and a ∨ u = a ∨ b = b ∨ v. The proof uses nothing more than the fact that each element in a regular σ-frame is a join of countably many elements each rather below it. Proposition 3.4. The following are equivalent for a completely regular locale L. (1) (2) (3) (4)
L is an Oz-locale. Every regular element of L is δ-regular. Every regular-closed sublocale of L is a regular Gδ -sublocale. Every regular-closed sublocale of L is a Gδ -coz-sublocale, and is completely separated from every zero-set sublocale disjoint from it. (5) RL is an Oz-ring. (6) RL is a splitting ring.
Proof. (1) ⇔ (2): If L is an Oz-locale, then any regular element is a cozero element, and hence it is a join of countably many elements completely below it. Since completely below implies rather below, the forward implication follows. Conversely, let a be a regular element in L and take a sequence (an ) in L such that an ≺ a for every n, and a = an . We may assume that each an is regular since, for any x and y in L, x ≺ y implies x∗∗ ≺ y.
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Now, exactly the same proof employed by Banaschewski and Gilmour to prove [5, Lemma 1] (the result quoted above just prior to this proposition) shows that L is mildly normal. Consequently, a ∈ Coz L, and therefore L is an Oz-locale. (2) ⇔ (3): This is proved in [17, p. 790]. (1) ⇒ (4): If L is an Oz-locale, then any regular a ∈ L is a cozero element, and so the second part of (4) follows from the normality of Coz L as a σ-frame. For the first part, take countably many an ∈ Coz L such that each an ≺≺ a and a = an . By Corollary 3 in [6], for each n we can find cn ∈ Coz L such that ∞ an ∧ cn = 0 and cn ∨ a = 1. We show that c(a) = o(cn ). Since a ∨ cn = 1, we have c(a) ⊆ o(cn ), which implies c(a) ⊆
∞
n=1
o(cn ). Since cn ≤
n=1
a∗n ,
we have
o(cn ) ⊆ o(a∗n ) ⊆ o(a∗n ) = c(a∗∗ n ), as a consequence of which c(a) ⊆
∞
o(cn ) ⊆
n=1
Therefore c(a) =
∞
∞
a∗∗ = c(a). n
c(a∗∗ n )=c
n=1
o(cn ).
n=1
(4) ⇒ (2): Let a be a regular element in L, and find countably many cozero elements cn such that ∞ c(a) = o(cn ). Then, for every n, c(a) ⊆ o(cn ), which implies a ∨ cn = 1. By the second part of the present n=1
hypothesis, c(a) is completely separated from c(cn ), hence we can find a cozero element dn such that dn ≺ a and dn ∨ cn = 1. Since Coz L is a normal σ-frame, we can thus find, for each n, un ≺ cn such that a ∨ un = 1. This of course implies u∗n ≺ a. Now,
c
∞ ∞ ∞ u∗n = o(un ) ⊆ c(u∗n ) = o(cn ) = c(a), n=1
n=1
n=1
which implies a ≤ u∗n , hence a = u∗n . Therefore a is δ-regular. (1) ⇒ (5): Suppose L is an Oz-locale, and let Q be an annihilator ideal in RL. Then there is an a ∈ L such that Q = M a∗ . Since L is an Oz-locale, a∗ ∈ Coz L, hence Q is a principal z-ideal, by Lemma 3.3, and therefore RL is an Oz-ring. (5) ⇒ (6): This is immediate. (6) ⇒ (1): We apply [4, Proposition 2.2(e)]. Let a ∧ b = 0 in L. Then a∗∗ ∧ b∗∗ = 0, and so M a∗∗ and M b∗∗ are annihilator ideals in RL with M a∗∗ ∩ M b∗∗ = {0}. By (6), there exist α, β ∈ RL such that M (α) ∩ M (β) = {0}, M a∗∗ ⊆ M (α) and M b∗∗ ⊆ M (β). The equality M (α) ∩ M (β) = {0} implies αβ = 0, whence coz α ∧ coz β = 0. Since a ≤ a∗∗ ≤ coz α and b ≤ b∗∗ ≤ coz β, it follows from [4, Proposition 2.2(e)] that L is an Oz-locale. 2 Remark 3.5. The equivalence (1) ⇔ (3) brings the following to mind. In [15, Proposition 3.5] it is proved that, in normal locales, c(a) is a Gδ -sublocale if and only if a is a cozero element. Thus, a normal locale is an Oz-locale if and only if every regular-closed sublocale is a Gδ -sublocale. If, analogously with spaces, we say a locale L is a Moscow locale in case every regular-closed sublocale is a join of Gδ -sublocales, then the equivalence (1) ⇔ (4) shows that every Oz-locale is a Moscow locale. Of course the converse is known to fail already in spaces (see [1, p. 385], for instance).
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Remark 3.6. The equivalence (1) ⇔ (2) is a variant of [4, Proposition 2.3]. It is observed in [4] that a dense quotient of an Oz-locale is an Oz-locale. Of course if a locale has a dense quotient (even a dense C ∗ -quotient) which is an Oz-locale, it does not follow that the locale is itself an Oz-locale. From the equivalence (1) ⇔ (5) we deduce that if h : L → M is a dense C-quotient map, then L is an Oz-locale if and only if M is an Oz-locale since the ring homomorphism Rh : RL → RM is an isomorphism. We shall now give a characterisation in terms of normal lower semicontinuous functions on a locale. This result extends Lane’s [19, Theorem 1.1 (iii)]. We thank Javier Gutiérrez García and Jorge Picado for showing us that it holds in locales, and also for supplying the proof. It is with their kind permission that we include the result here. We need some background, but we shall be terse about it and refer to the articles [16] and [17] for more details. Let S(L) denote the locale S(L)opp and L(R) be the locale of reals. The ring of arbitrary real functions on L, denoted F(L), is defined in [14] to be the ring R(S(L)). The subring of F(L) consisting of bounded functions is denoted by F∗ (L). In [17] the authors define what it means for an f ∈ F(L) to be normal lower semicontinuous. The set of bounded normal lower semicontinuous functions on L is denoted by NLSC∗ (L). In [16, Example 4.6], the authors define the characteristic function, χS , for each complemented S ∈ S(L) in the following manner. Recall that scales can be used to define localic real functions. The function σ : Q → S(L) defined by ⎧ ⎪ ⎪ ⎨1S(L) σ(p) = S ∗ ⎪ ⎪ ⎩0 S(L)
if p < 0 if 0 ≤ p < 1 if p ≥ 1
is a scale on S(L), and χS ∈ F∗ (L) is the real function it defines. Explicitly, χS maps the generators of the form (p, −) as follows: ⎧ ⎪ ⎪ ⎨1S(L) χS (p, −) = S ∗ ⎪ ⎪ ⎩0 S(L)
if p < 0 if 0 ≤ p < 1 if p ≥ 1.
It is observed in [16, Remark 4.7] that, for any a ∈ L, χo(a) ∈ NLSC∗ (L) if and only if a = a∗∗ . Proposition 3.7 (Gutiérrez García and Picado). A completely regular locale L is an Oz-locale iff for any f ∈ NLSC∗ (L), f (p, −) is a zero-set sublocale for every p ∈ Q. Proof. (⇒) Assume L is an Oz-locale. Let f ∈ NLSC∗ (L) and p ∈ Q. Now, Corollary 3.5 in [17] implies that f (p, −) = r>p f (r, −)◦ . Observe that this is a closed sublocale since the join is contemplated in S(L). Denoting this sublocale by c(ap ), the fact that f is normal lower semicontinuous implies ap = r>p a∗∗ r , as shown in [17, Corollary 3.5]. Since L is an Oz-locale, f (p, −) is a zero-set sublocale since r>p a∗∗ is then a r join of countably many cozero elements, so that it is itself a cozero element. (⇐) Assume the condition holds, and let a ∈ L be regular. By the remark above cited from [16], we have that χo(a) ∈ NLSC∗ (L), and so, by the current hypothesis, the closed sublocale c(a) = χo(a) (0, −) is a zero-set sublocale, which means that a ∈ Coz L. Therefore L is an Oz-locale. 2 To close this section, let us recall from [4] that a locale L is called a weak Oz-locale if a∗ ∈ Coz L for every a ∈ Coz L. Two characterisations of these locales are given in [4] including one that says L is a weak
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Oz-locale if and only if for any a ∈ L and c ∈ Coz L such that a ∧ c = 0, there exists d ∈ Coz L such that a ≤ d and c ∧ d = 0 [4, Proposition 5.3]. Here we give some ring-theoretic characterisations. Definition 3.8. We say a ring A is a weak Oz-ring if every element annihilator of A is a principal z-ideal; and we say A is a weak splitting ring if for any element annihilator I and any principal z-ideal U with I ∩ U = 0, there exists a principal z-ideal V such that U ∩ V = 0 and I ⊆ V . A proof along the lines of the equivalences of (1), (5) and (6) in Proposition 3.4 above establishes the following. Proposition 3.9. The following are equivalent for a completely regular locale L. (1) L is a weak Oz-locale. (2) RL is a weak Oz-ring. (3) RL is a weak splitting ring. 4. When βL is an Oz-locale In this section we shall give some necessary and sufficient conditions for βL to be an Oz-locale under some (necessary) assumption on L. The assumption is the following weakening of normality. In his study of countable paracompactness and weak normality conditions, Mack [20] introduces what he calls weakly δ-normally separated spaces. These are spaces in which every regular-closed set is completely separated from every zero-set disjoint from it. Locales with these features were considered in [13] and were named “weakly δ-normally separated locales”. This is quite a mouthful, so here we shall abbreviate and say such locales are wδ-normal. Thus, paraphrasing [13, Lemma 3.14], we have that L is wδ-normal iff for every regular a ∈ L and c ∈ Coz L with a ∨ c = 1, there exists d ∈ Coz L such that d ≤ a and d ∨ c = 1. By the normality of the σ-frame Coz L, if L is wδ-normal and a ∨ c = 1 for some regular a ∈ L and c ∈ Coz L, then there exists b ∈ Coz L such that b ≺≺ a and b ∨ c = 1. One upshot of this is that we then have rL (a) ∨ rL (c) = 1βL because rL preserves finite joins of cozero elements. Trivially, every Oz-locale is wδ-normal. Lemma 4.1. Let L be a completely regular wδ-normal locale. For any regular a ∈ L, rL (a) ∈ Coz(βL) iff c(a) has a countable cozero neighbourhood base for its cozero neighbourhoods. Proof. (⇒) Assume rL (a) ∈ Coz(βL). By [6, Corollary 4], rL (a) is a Lindelöf element in βL. Since rL (a) =
{rL (t) | t ≺≺ a},
we can find an increasing sequence (tn ) of cozero elements of L such that tn ≺≺ a for each n, and rL (a) =
{rL (tn ) | n ∈ N}.
Since rL (tn ) ≺≺ rL (a) for each n, we can find In ∈ Coz(βL) such that rL (tn ) ∧ In = 0βL
and
In ∨ rL (a) = 1βL .
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Put cn = In , and observe that each cn is a cozero element of L with a ∨ cn = 1. That is, each o(cn ) is a cozero neighbourhood of c(a). Now let o(d), with d ∈ Coz L, be any cozero neighbourhood of c(a). Then a ∨ d = 1, hence, as remarked just prior to the statement of the lemma, rL (a) ∨ rL (d) = 1βL , that is, rL (tn ) ∨ rL (d) = 1βL . n
By compactness of βL, and because the sequence (tn ) is increasing, there is an n such that rL (tn ) ∨ rL (d) = 1βL . Since In ∧ rL (tn ) = 0βL , we have In ≤ rL (d), whence cn ≤ d. Therefore o(cn ) ⊆ o(d), which establishes the implication. (⇐) Assume the stated condition holds, and find countably many cn ∈ Coz L such that {o(cn ) | n ∈ N} is a base for the cozero neighbourhoods of c(a). Then a ∨ cn = 1 for every n. Since L is wδ-normal and a is regular, we can find, for each n, an element un ∈ Coz L such that un ≺≺ a and un ∨ cn = 1. We claim that rL (a) =
rL (un ).
n
The containment
rL (un ) ⊆ rL (a) is immediate. To show the other containment, let t ≺≺ a, and pick
n
w ∈ Coz L with t ≺≺ w ≺≺ a. Next, take s ∈ Coz L such that w ∧ s = 0 and s ∨ a = 1. Then o(s) is a cozero neighbourhood of c(a), hence, by the present hypothesis, there is an m such that cm ≤ s. Then w ∧ cm = 0, and hence w ≺≺ um (with cm as a separating element), which implies t ≺≺ um , so that t ∈ rL (um ). Thus rL (a) ⊆
n
rL (un ) ⊆
rL (un ),
n
and hence equality. Since rL (un ) ≺≺ rL (a) for each n, it follows that rL (a) ∈ Coz(βL).
2
Let us observe the following. If h : M → L is a dense frame homomorphism, then h∗ h(a) = a for every regular a ∈ M . Indeed, the equality h h∗ h(a) ∧ a∗ = 0 implies h∗ h(a) ∧ a∗ = 0 by density of h, whence h∗ h(a) ≤ a∗∗ = a, and hence the claimed statement. Proposition 4.2. The following are equivalent for a wδ-normal locale L. (1) βL is an Oz-locale. (2) Every regular-closed sublocale of L has a countable cozero neighbourhood base for its cozero neighbourhoods. Proof. (1) ⇒ (2): Let a ∈ L be regular. Then rL (a) is a regular element in βL, and so rL (a) ∈ Coz(βL) since βL is an Oz-locale. Therefore, by Lemma 4.1, c(a) has a countable cozero neighbourhood base for its cozero neighbourhoods. (2) ⇒ (1): Let I be a regular element of βL, and write a = I. Then a is a regular element of L, and I = rL (a) by what we observed above. By the current hypothesis, c(a) has a countable cozero neighbourhood base for its cozero neighbourhoods, which then makes rL (a) a cozero element in βL by Lemma 4.1. Therefore βL is an Oz-locale. 2 For our next characterisation we need some terminology, notation, and three lemmas. First, following [9], we say a sublocale S of a locale L is peripheral if it is of the form c(a ∨ a∗ ), for some regular a ∈ L. Next, for any a ∈ L we write κa : L → ↑a for the frame homomorphism x → a ∨ x. We present the first lemma in frame language. Let us therefore recall complete separation of quotients β α in frame terms. Let A ←− L −→ B be quotients of L, which is to say α and β are surjective frame
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homomorphisms. The quotients A and B are completely separated if there exist c, d ∈ Coz L with c ∨ d = 1 such that α(c) = 0A and β(d) = 0B . Note that for the homomorphism κa : L → ↑a, κ(c) = 0↑a if and only if c ≤ a. Recall that if u ∨ v = 1 in Coz L, then there exists w ∈ Coz L such that w ≺≺ v and u ∨ w = 1. h
Lemma 4.3. Let L −→ M be a quotient of L. Suppose that there is a sequence (an) in L such that, for every c ∈ Coz L with h(c) = 1, there is an index m with am ≤ c. Suppose further that, for each n, the quotients κa
h
n ↑an ←− L −→ M are completely separated. Then rL h∗ (0) ∈ Coz(βL).
κa
h
un ∨ vn = 1,
and
n Proof. By complete separation of the quotients ↑an ←− L −→ M , there exist, for each n, cozero elements of un and vn of L such that
un ≤ an ,
h(vn ) = 0.
Then vn ≤ h∗ (0). Now take wn ∈ Coz L such that wn ≺≺ vn and un ∨ wn = 1. We claim that rL h∗ (0) = The containment
{rL (wn ) | n ∈ N}.
rL (wn ) ⊆ rL h∗ (0) is immediate because wn ≤ h∗ (0) for every n. Now let t ≺≺ h∗ (0), n
and pick s ∈ Coz L such that t ∧ s = 0 and s ∨ h∗ (0) = 1. Then h(s) = 1, and so, by hypothesis, there is an index m such that am ≤ s. Now, t ∧ u m ≤ t ∧ am ≤ t ∧ s = 0
and
um ∨ wm = 1,
from which we can deduce that t ≺≺ wm because the separating element um is a cozero element in L. Therefore t ∈ rL (wn ), so that rL h∗ (0) ⊆ rL (wn ), and hence equality. Since rL (wn ) ≺≺ rL h∗ (0) for n
n
each n, it follows that rL h∗ (0) is a cozero element in βL. 2 Lemma 4.4. Let a and b be regular elements in a locale L. If a ∨ a∗ ∨ b = 1, then a∗ ∨ b is regular. Proof. Since a ∨ (a∗ ∨ b) = 1, (a∗ ∨ b)∗∗ = (a∗ ∨ b)∗∗ ∧ a ∨ (a∗ ∨ b)∗∗ ∧ (a∗ ∨ b) = (a∗ ∨ b)∗∗ ∧ a∗∗ ∨ (a∗ ∨ b) ∗∗ = (a∗ ∨ b) ∧ a ∨ (a∗ ∨ b) = (a ∧ b)∗∗ ∨ (a∗ ∨ b) = (a ∧ b) ∨ (a∗ ∨ b)
since a and b are regular
= (a∗ ∨ b), which proves the result. 2 Finally, we shall also need the following lemma which appears as Proposition 3.3.6 in [23]. We shall give only an outline of the proof. Lemma 4.5. Let L be a mildly normal frame. If a ∨ b = 1 where a and b are regular elements of L, then the κb κa quotients ↑a ←− L −→ ↑b are completely separated.
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Proof. One shows first that if s and t are regular elements in L with s ≺ t, then for some regular r ∈ L we have s ≺ r ≺ t. The proof mimics the one that shows that the rather below relation interpolates in normal frames. Here, the rather below relation between regular elements interpolates through a regular element. Thus, by Countable Dependent Choice, s ≺ t implies s ≺≺ t. Then the proof of [2, Proposition 8.3.1] is copied verbatim (except that one restricts to regular elements) to show that there exist c, d ∈ Coz L such that c ≤ a, d ≤ b and c ∨ d = 1. This establishes the complete separation of the quotients. 2 An immediate consequence of this is that if L is mildly normal and a, b ∈ L are regular, then rL (a) ∨ rL (b) = 1βL . Proposition 4.6. The following are equivalent for a mildly normal completely regular locale L. (1) βL is an Oz-locale. (2) Every regular-closed sublocale of L has a countable regular-open neighbourhood base for its regular-open neighbourhoods. (3) Every peripheral sublocale of L has a countable regular-open neighbourhood base for its regular-open neighbourhoods. Proof. (1) ⇒ (2): Let a be a regular element in L. Then rL (a) is regular in βL, and is therefore a cozero element because βL is an Oz-locale. Thus, there are elements In in βL such that In ≺≺ In+1 for every n, and rL (a) = In . For each n choose a regular Jn in βL such that n
In ∧ Jn = 0βL
and
rL (a) ∨ Jn = 1βL .
For each n let bn = Jn , and note that each bn is a regular element of L. Also, applying the map jL : βL → L to the equality rL (a) ∨ Jn = 1βL , we see that a ∨ bn = 1 for each n, so that each o(bn ) is a neighbourhood of c(a). Let x be a regular element of L with a ∨ x = 1. Since L is mildly normal, rL (a) ∨ rL (x) = 1βL . That is, rL (x) ∨ In = 1βL , which, by the compactness of βL, and the fact that the sequence (In ) is increasing, n
implies that there is an index k such that rL (x) ∨ Ik = 1βL . Since Ik ∧ Jk = 0βL , it follows that Jk ≤ rL (x), and hence bk = Jk ≤ rL (x) = x, so that o(bk ) ⊆ o(x). Therefore the set {o(bn ) | n ∈ N} is a countable regular-open neighbourhood base for the regular-open neighbourhoods of c(a). (2) ⇒ (1): Let I be a regular element of βL. Then I = rL (a) for the regular element a = I of L. By the current hypothesis, there are regular elements un in L such that the collection {o(un ) | n ∈ N} is a neighbourhood base for the regular-open neighbourhoods of c(a). Observe that, for each n, the quotients κun κa ↑un ←− L −→ ↑a are completely separated because a ∨ un = 1 and L is mildly normal. Furthermore, if κa (u) = 1 for some u ∈ L, then a ∨ u = 1, and hence un ≤ u for some n. Therefore, by Lemma 4.3, rL (κa )∗ (0↑a ) ∈ Coz(βL). But rL (κa )∗ (0↑a ) = rL (a), so we are done. (2) ⇔ (3): Assume (2) and consider any peripheral sublocale c(a ∨ a∗ ) of L, with a regular. Let {o(un ) | n ∈ N} and {o(vn ) | n ∈ N} be regular-open neighbourhood bases for the regular-open neighbourhoods of c(a) and c(a∗ ), respectively, which are guaranteed by (2). Let o(x) be a regular-open neighbourhood of c(a ∨a∗ ). By Lemma 4.4, a ∨x and a∗ ∨x are regular elements, hence o(a∗ ∨x) is a regular-open neighbourhood of c(a), and o(a ∨ x) is a regular-open neighbourhood of c(a∗ ). Take k, ∈ N such that o(uk ) ⊆ o(a∗ ∨ x) and o(v ) ⊆ o(a ∨ x). A simple calculation shows that o(uk ∧ v ) ⊆ o(x). This shows that the collection {o(um ∧ vn ) | m, n ∈ N} is a countable regular-open neighbourhood base for regular-open neighbourhoods of c(a ∨ a∗ ). Conversely, assume (3) holds and let c(a) be a regular-closed sublocale of L. Let {o(wn ) | n ∈ N} be a regular-open neighbourhood base for the regular-open neighbourhoods of c(a ∨ a∗ ). By Lemma 4.4, each o(a∗ ∨ wn ) is a regular-open neighbourhood of c(a). Let o(x) be a regular-open neighbourhood of c(a). Then
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o(x) is also a regular-open neighbourhood of c(a ∨ a∗ ), and so there exists m ∈ N such that o(wm ) ⊆ o(x). Since a ∨ x = 1 (as c(a) ⊆ o(x)), a∗ ≤ x, which then implies a∗ ∨ wn ≤ x, whence o(a∗ ∨ wm ) ⊆ o(x). We then deduce that {o(a∗ ∨ wn ) | n ∈ N} is a countable regular-open neighbourhood base for regular-open neighbourhoods of c(a). 2 Borrowing terminology from spaces [25], we say a locale L is almost normal if every regular-closed sublocale of L is completely separated from every closed sublocale disjoint from it. It is easy to show that if L is almost normal and a ∨ b = 1 with a regular, then there is a c ≺ b such that a ∨ c = 1. We therefore have the following corollary. Corollary 4.7. The following are equivalent for an almost normal locale L. (1) βL is an Oz-locale. (2) Every regular-closed sublocale of L has a countable basis for its neighbourhoods. Let us recall that λL is always a normal locale and β(λL) ∼ = βL. We can thus put together the various characterisations above of when βL is an Oz-locale into a single corollary stated in terms of λL as follows. Corollary 4.8. The following are equivalent for any completely regular locale L. (1) βL is an Oz-locale. (2) Every regular-closed sublocale of λL has a countable cozero neighbourhood base for its cozero neighbourhoods. (3) Every regular-closed sublocale of λL has a countable regular-open neighbourhood base for its regular-open neighbourhoods. (4) Every regular-closed sublocale of λL has a countable basis for its neighbourhoods. (5) Every peripheral sublocale of λL has a countable regular-open neighbourhood base for its regular-open neighbourhoods. In closing, let us consider when βL is a weak Oz-locale. Observe that if L is a weak Oz-locale, then for any a ∈ Coz L, rL (a∗ ) ∨ rL (c) = 1βL whenever a and c are cozero elements with a∗ ∨ c = 1. In the proof of Lemma 4.1 the fact that L is a wδ-normal locale was used to deduce that rL (a) ∨ rL (c) = 1βL whenever a ∨ c = 1 for regular a and c ∈ Coz L. Hence, the same proof yields the following lemma. A support of L is a closed sublocale of the form c(a∗ ), for a ∈ Coz L. This term is of course borrowed from classical topology and agrees with its topological namesake because c(a∗ ) = o(a). Lemma 4.9. Let L be a weak Oz-locale. For any a ∈ Coz L, rL (a∗ ) ∈ Coz(βL) iff c(a∗ ) has a countable cozero neighbourhood base for its cozero neighbourhoods. Proposition 4.10. The following are equivalent for a weak Oz-locale L. (1) βL is a weak Oz-locale. (2) Every support of L has a countable cozero neighbourhood base for its cozero neighbourhoods. Proof. Assume βL is a weak Oz-locale and consider any support c(a∗ ) of L with a ∈ Coz L. Since jL : βL → L is coz-onto, there exists I ∈ Coz(βL) such that I = a. Then rL (a∗ ) = I ∗ ∈ Coz(βL) since βL is a weak Oz-locale. Therefore c(a∗ ) has the stated property by the foregoing lemma.
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Conversely, suppose the stated condition holds. Let I ∈ Coz(βL) and put a = I. Then a ∈ Coz L, and hence a∗ ∈ Coz L since L is a weak Oz-locale. Now, I ∗ = rL (a∗ ), whence I ∗ ∈ Coz(βL) by the lemma. Therefore βL is a weak Oz-locale. 2 There are also characterisations for when βL is a weak Oz-locale analogous to those in Proposition 4.6 under the hypothesis that L is a weak Oz-locale. Let us say a sublocale of L is a co-support if it is of the form o(c∗ ), with c ∈ Coz L; and that it is coz-peripheral if it is of the form c(c∗ ∨ c∗∗ ), for some c ∈ Coz L. Observe that if L is a weak Oz-locale, then c∗∗ ∈ Coz L for every c ∈ Coz L, hence, by Lemma 4.4, for any d ∈ Coz L with c∗ ∨ c∗∗ ∨ d∗ = 1, c∗∗ ∨ d∗ is a pseudocomplement of some cozero element of L. Using this observation (as a “weak Oz” analogue of Lemma 4.4), we can prove along similar lines to Proposition 4.6 the following. Proposition 4.11. The following are equivalent for a weak Oz-locale L. (1) βL is a weak Oz-locale. (2) Every support of L has a countable coz-support neighbourhood base for its coz-support neighbourhoods. (3) Every coz-peripheral sublocale of L has a countable coz-support neighbourhood base for its coz-support neighbourhoods. Acknowledgements Thanks are due to the referee for detailed comments and helpful suggestions which have improved this paper. References [1] A.V. Arhangel’skii, Moscow spaces and topological groups, Topol. Proc. 25 (2000) 383–416. [2] R.N. Ball, J. Walters-Wayland, C- and C ∗ -quotients in pointfree topology, Diss. Math. (Rozprawy Mat.) 412 (2002), 62 pp. [3] B. Banaschewski, The real numbers in pointfree topology, Textos de Matemática Série B, vol. 12, Departamento de Matemática da Universidade de Coimbra, 1997. [4] B. Banaschewski, T. Dube, C. Gilmour, J. Walters-Wayland, Oz in pointfree topology, Quaest. Math. 32 (2009) 215–227. [5] B. Banaschewski, C. Gilmour, Stone–Čech compactification and dimension theory for regular σ-frames, J. Lond. Math. Soc. 39 (1985) 1–8. [6] B. Banaschewski, C. Gilmour, Pseudocompactness and the cozero part of a frame, Comment. Math. Univ. Carol. 37 (1996) 577–587. [7] B. Banaschewski, C. Gilmour, Oz revisited, in: H. Herrlich, H.-E. Porst (Eds.), Proceedings of the Conference on Categorical Methods in Algebra and Topology, in: Math. Artbeitspapiere, vol. 54, Universität Bremen, 2000, pp. 19–23. [8] R.L. Blair, Spaces in which special sets are z-embedded, Can. J. Math. 28 (4) (1976) 673–690. [9] R.L. Blair, M.A. Swardson, Spaces with an Oz Stone–Čech compactification, Topol. Appl. 36 (1990) 73–92. [10] T. Dube, Some ring-theoretic properties of almost P -frames, Algebra Univers. 60 (2009) 145–162. [11] T. Dube, Contracting the socle in rings of continuous functions, Rend. Semin. Mat. Univ. Padova 123 (2010) 37–53. [12] T. Dube, O. Ighedo, On z-ideals of pointfree function rings, Bull. Iran. Math. Soc. 40 (2014) 655–673. [13] T. Dube, M. Matlabyana, Concerning variants of C-embedding in pointfree topology, Topol. Appl. 158 (2011) 2307–2321. [14] J. Gutiérrez García, T. Kubiak, J. Picado, Localic real functions: a general setting, J. Pure Appl. Algebra 213 (2009) 1064–1074. [15] J. Gutiérrez García, T. Kubiak, J. Picado, Pointfree forms of Dowker’s and Michael’s insertion theorems, J. Pure Appl. Algebra 213 (2009) 98–108. [16] J. Gutiérrez García, I. Mozo Carolo, J. Picado, Normal semicontinuity and the order completion of pointfree function rings (Preprint). [17] J. Gutiérrez García, J. Picado, On the parallel between normality and extremal disconnectedness, J. Pure Appl. Algebra 218 (2014) 784–803. [18] P.B. Johnson, κ-Lindelöf locales and their spatial parts, Cah. Topol. Géom. Différ. Catég. 32 (1991) 297–313. [19] E.P. Lane, PM-normality and the insertion of a continuous function, Pac. J. Math. 82 (1979) 155–162. [20] J. Mack, Countable paracompactness and weak normality properties, Trans. Am. Math. Soc. 148 (1970) 265–272. [21] J. Madden, J. Vermeer, Lindelöf locales and realcompactness, Math. Proc. Camb. Philos. Soc. 99 (1986) 473–480. [22] G. Mason, z-Ideals and prime ideals, J. Algebra 26 (1973) 280–297.
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M.Z. Matlabyana, Coz-related and other special quotients in frames, PhD thesis, Univ. South Africa, 2012. J. Picado, A. Pultr, Frames and Locales: Topology without Points, Frontiers in Mathematics, Springer, Basel, 2012. M.K. Singal, S.P. Arya, Almost normal and almost completely regular spaces, Glas. Mat. Ser. 3 5 (25) (1970) 145–152. M.K. Singal, A.R. Singal, Mildy normal spaces, Kyungpook Math. J. 1 (1973) 27–31. T. Terada, On spaces whose Stone–Čech compactification is Oz, Pac. J. Math. 85 (1979) 231–237.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
More on locales in which every open sublocale is z-embedded ✩ Themba Dube ∗ , Oghenetega Ighedo Department of Mathematical Sciences, University of South Africa, P.O. Box 392, 0003 Unisa, South Africa
a r t i c l e
i n f o
Article history: Received 31 December 2014 Accepted 8 June 2015 Available online 22 December 2015 MSC: 06D22 54C25 54C30 54D15
a b s t r a c t The locales which have the feature stated in the title (we shall call them Ozlocales) were introduced by Banaschewski and Gilmour as localic extensions of Blair’s Oz-spaces. In this note we give further characterisations of Oz-locales and their generalisations which we shall call weak Oz-locales. These characterisations include ring-theoretic ones which are new even for spaces. Among locales L which satisfy some weaker forms of normality, we characterise those for which βL is an Oz-locale. © 2015 Elsevier B.V. All rights reserved.
Keywords: Locale Oz-locale Sublocale Stone–Čech compactification Principal z-ideal
1. Introduction A Tychonoff space in which every open set is z-embedded is called an Oz-space. These spaces were introduced by Blair [8] who gave several characterisations of them. Subsequently they were also studied by other authors such as Lane [19] and Terada [27]; the latter concentrating mainly on characterising those Oz-spaces whose Stone–Čech compactifications are also Oz-spaces. Oz-spaces are also called perfectly κ-normal by some authors. On the other hand, Oz-locales (which generalise Oz-spaces in a natural and conservative way) were introduced by Banaschewski and Gilmour [7], and studied further in [4] where they are characterised in several ways. We shall here call them Oz-locales to be in line with such concepts as sublocale, Gδ -sublocale, and so on, which shall feature prominently in this discussion. A thorough literature review of Oz-spaces and ✩ This work is based on the research supported in part by the National Research Foundation of South Africa (Grant Number 93514). * Corresponding author. E-mail address: [email protected] (T. Dube).
http://dx.doi.org/10.1016/j.topol.2015.12.030 0166-8641/© 2015 Elsevier B.V. All rights reserved.
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Oz-locales shows that nowhere are these objects characterised in terms of their rings of continuous functions. One of our goals in this note is to give more characterisations of Oz-locales, including ring-theoretic ones which are new even for spaces. Among locales L which satisfy some weaker forms of normality, we characterise for which βL is an Oz-locale. These weaker forms of normality are localic versions of what Mack [20] calls weak δ-normality, almost normality introduced by Singal and Arya [25], and mild normality introduced by Singal and Singal [26]. Here is a brief outlay of the paper. We collect a few reminders in Section 2, concentrating mainly on those concepts which are not included in our main reference [24]. Following that, we present characterisations of Oz-locales and weak Oz-locales in Section 3, and in the last section we characterise when the Stone–Čech compactification is an Oz-locale. As mentioned above, we have put some normality condition on a locale L to obtain characterisations for βL to be an Oz-locale. We should remark though that, for an arbitrary L, if βL is an Oz-locale, then L does satisfy all the normality conditions we have imposed. In one of the corollaries we give necessary and sufficient conditions in terms of properties of λL (the regular Lindelöf reflection of L in the category Loc) for βL to be an Oz-locale. These of course do not have topological analogues. 2. Preliminaries 2.1. Frames Our reference for the general theory of frames and locales is [24]. Throughout the paper all frames are assumed to be completely regular. In certain instances we will mention complete regularity for emphasis. Let L be a frame. We view the Stone–Čech compactification of L as the frame of completely regular ideals of L. We denote by jL : βL → L the join map J → J. The right adjoint of jL is here denoted by rL . Recall that, for any a ∈ L, rL (a) = {x ∈ L | x ≺≺ a}. We write, as usual, Coz L for the set of cozero elements of L. If a, b ∈ Coz L, then rL (a ∨ b) = rL (a) ∨ rL (b); and if L is normal, then rL (x ∨ y) = rL (x) ∨ rL (y) for all x, y ∈ L. We shall frequently use the fact that Coz L is a normal σ-frame, as a consequence of which (see [6]) we have that whenever c ∨ d = 1 in Coz L, then there exist u, v ∈ Coz L such that u ≺≺ c, v ≺≺ d and u ∨ v = 1. A frame homomorphism h : L → M is coz-onto if for every d ∈ Coz M there is a c ∈ Coz L such that h(c) = d. By a point of L we mean a prime element, that is, an element p < 1 such that for any a and b in L, a ∧ b ≤ p implies a ≤ p or b ≤ p. We denote by Pt(L) the set of all points of L. We remark that, subject to appropriate choice principles (which we assume throughout), a compact regular frame has enough points, which means that every element is a meet of points. An element a ∈ L is called regular if a = a∗∗ . It is dense if a∗ = 0. A frame homomorphism is dense if the bottom element of its domain is the only element it maps to the bottom of its codomain. If h : M → L is dense onto, then (i) for any a ∈ M , h(a∗ ) = h(a)∗ , and (ii) for any b ∈ L, h∗ (b∗ ) = (h∗ (b))∗ . A consequence of this is that dense onto homomorphisms send regular elements to regular elements; as do their right adjoints. 2.2. The Lindelöf coreflection An ideal of Coz L is a σ-ideal if it is closed under countable joins. The regular Lindelöf coreflection of L (see [21] for details), denoted λL, is the frame of σ-ideals of Coz L. The join map λL : λL → L is a dense onto
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frame homomorphism, and is the coreflection map to L from completely regular Lindelöf frames. For any a ∈ L we write JaK = {c ∈ Coz L | c ≤ a}, and recall that the right adjoint of λL is given by (λL )∗ (a) = JaK. The map ηL : βL → λL
ηL (I) = I σ ,
given by
where · σ signifies σ-ideal generation in Coz L, is a dense onto frame homomorphism. In fact, ηL : βL → λL ηL
λ
L L is exactly realises the Stone–Čech compactification of λL. Furthermore, the composite βL −→ λL −→ the map jL : βL → L.
2.3. Sublocales Let L be a locale. As in [24], S(L) designates the lattice of all sublocales of L. We use the notation o(a) and c(a), respectively, for open and closed sublocales of L determined by a ∈ L. If a is regular we say o(a) is regular-open and c(a) is regular-closed. If a ∈ Coz L, we say o(a) is a cozero sublocale and c(a) is a zero-set sublocale; the latter term comes from [18]. Following [17], we say an element of L is δ-regular if it is the join of countably many elements each rather below it. Cozero elements are δ-regular. Following [17] again, but this time upside down, we say a sublocale S is a regular Gδ -sublocale if it is the intersection of countably many of its closed neighbourhoods. In [17] this is stated “dually” because the authors are working with the frame S(L)opp and not the co-frame S(L). We have thus used the hyphen to distinguish ∞ between the two terminologies. By a Gδ -coz-sublocale of L we mean a sublocale of the form o(cn ) where each cn ∈ Coz L. Of course a Gδ -sublocale is one of the form
∞
n=1
o(an ) where the an are arbitrary elements
n=1
in L. By a neighbourhood of a sublocale S ⊆ L we mean a sublocale T ⊆ L such that S ⊆ o(a) ⊆ T for some a ∈ L. That is, a sublocale whose interior contains S. A cozero neighbourhood of S is any o(a) ⊇ S with a ∈ Coz L. We say a collection G ⊆ S(L) is a neighbourhood base for S if each member of G is neighbourhood of S and every neighbourhood of S contains a member of G. The closure of a sublocale S will be denoted by S. The reader will recall that S
S=↑
and
o(a) = c(a∗ ).
The smallest sublocale of L is denoted by O. We say sublocales S and T are disjoint if S ∩ T = O. 2.4. Function rings Our approach to pointfree function rings is that of [3]. For any I ∈ βL, the sets M I = {α ∈ RL | rL (coz α) ⊆ I}
and
O I = {α ∈ RL | rL (coz α) ≺ I}
are ideals in the ring RL. For any a ∈ L we abbreviate M rL (a) as M a , and remark that M a = {α ∈ RL | coz α ≤ a}. Observe that if M a = M b for any a, b ∈ L, then a = b by complete regularity. The maximal ideals of RL are precisely the ideals M I , for I ∈ Pt(βL) (see [10]). The annihilator ideals in RL are precisely the ideals M a∗ , for a ∈ L [11, Lemma 3.1]. In particular, for any α ∈ RL, Ann(α) = M (coz α)∗ . Call an ideal of a
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ring A an element annihilator if it is of the form Ann(a) for some a ∈ A. Element annihilators in RL are exactly the ideals M c∗ , for c ∈ Coz L. 2.5. z-Ideals Let A be a commutative ring with identity, and let Max(A) denote the set of maximal ideals of A. We set M(a) = {M ∈ Max(A) | a ∈ M }
and
M (a) =
M(a).
Recall from [22] that an ideal I of A is called a z-ideal if, for every a, b ∈ A, a ∈ I and M(a) = M(b)
=⇒
b ∈ I.
Every maximal ideal is a z-ideal, and intersections of z-ideals are z-ideals. 3. Some characterisations of Oz-locales Let us start by recalling the pertinent definitions. First, an element s ∈ L is called coz-embedded if the frame homomorphism L → ↓s, given by x → s ∧ x, is coz-onto. The locale L is an Oz-locale if every a ∈ L is coz-embedded. We shall frequently use one or the other of the various characterisations, which we now record in the following proposition. The proofs are in [7] and [4]. For a locale L, the symbol L∗ denotes the sub-σ-frame of L generated by regular elements of L. Proposition 3.1. The following are equivalent for a completely regular locale L. (1) (2) (3) (4) (5) (6)
L is an Oz-locale. Every dense element of L is coz-embedded. For every a ∈ L, a∗ ∈ Coz L. For every regular a ∈ L, a ∨ a∗ ∈ Coz L. For all disjoint a, b ∈ L, there are disjoint c, d ∈ Coz L such that a ≤ c and b ≤ d. L∗ is a regular σ-frame.
As mentioned in the abstract, some of the characterisations we shall present are ring-theoretic. So we first introduce some terminology. It may seem odd that the rings we are about to define we will call Oz-rings, but it will soon be apparent why such a moniker is appropriate. Definition 3.2. An ideal of a ring A is a principal z-ideal if it is of the form M (a) for some a ∈ A. We say a ring is (i) an Oz-ring if every annihilator ideal of the ring is a principal z-ideal. (ii) a splitting ring if for any annihilator ideals I and J with I ∩ J = 0, there exist principal z-ideals U and V with U ∩ V = 0 such that I ⊆ U and J ⊆ V . A simple example of an Oz-ring is any integral domain with zero Jacobson radical, such as the ring Z. The following lemma gives a description of principal z-ideals in RL. Lemma 3.3. The principal z-ideals of RL are precisely the ideals M c , for c ∈ Coz L.
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Proof. It suffices to show that, for any α ∈ RL, M (α) = M coz α . It is shown in the proof of [12, Lemma 3.2] that, for any I ∈ βL, MI =
{M J | J ∈ Pt(βL) and I ⊆ J}.
Since for any J ∈ Pt(βL) we have α ∈ M J if and only if rL (coz α) ⊆ J, it follows that M coz α = M
rL (coz α)
{M J | J ∈ Pt(βL) and rL (coz α) ⊆ J} = {M J | J ∈ Pt(βL) and α ∈ M J }
=
= M (α), which proves the result. 2 Recall that two sublocales S and T of a locale L are said to be completely separated [2] if they have disjoint zero-set sublocale neighbourhoods. In particular, for any a, b ∈ L, the sublocales c(a) and c(b) are completely separated if and only if there exist c, d ∈ Coz L such that c ∧ d = 0 and a ∨ c = 1 = b ∨ d. Thus, if the closed sublocales c(a) and c(b) are completely separated, then there exists d ∈ Coz L such that d ≺ a and d ∨ b = 1. Indeed, if c and d are cozero elements with c ∧ d = 0 and a ∨ c = 1 = b ∨ d, then d ≺ a (with c as a witness for this) and d ∨ b = 1. A locale L is called mildly normal [17] if whenever a, b are regular elements such that a ∨ b = 1, then there exist u, v ∈ L such that u ∧ v = 0 and a ∨ u = 1 = b ∨ v. In a mildly normal locale every δ-regular element is a cozero element (see Remark 3.2(4) in [17]). In fact, this holds in what are called almost normal locales which include the mildly normal ones. The proof of one implication in the following result is modelled on that of (i) ⇒ (v) of [19, Theorem 1.1]. Also, we call the attention of the reader to the proof Banaschewski and Gilmour of [5, Lemma 1] which states (we are not quoting it verbatim) that If a and b are elements of a regular σ-frame, then there exist u and v such that u ∧ v = 0 and a ∨ u = a ∨ b = b ∨ v. The proof uses nothing more than the fact that each element in a regular σ-frame is a join of countably many elements each rather below it. Proposition 3.4. The following are equivalent for a completely regular locale L. (1) (2) (3) (4)
L is an Oz-locale. Every regular element of L is δ-regular. Every regular-closed sublocale of L is a regular Gδ -sublocale. Every regular-closed sublocale of L is a Gδ -coz-sublocale, and is completely separated from every zero-set sublocale disjoint from it. (5) RL is an Oz-ring. (6) RL is a splitting ring.
Proof. (1) ⇔ (2): If L is an Oz-locale, then any regular element is a cozero element, and hence it is a join of countably many elements completely below it. Since completely below implies rather below, the forward implication follows. Conversely, let a be a regular element in L and take a sequence (an ) in L such that an ≺ a for every n, and a = an . We may assume that each an is regular since, for any x and y in L, x ≺ y implies x∗∗ ≺ y.
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Now, exactly the same proof employed by Banaschewski and Gilmour to prove [5, Lemma 1] (the result quoted above just prior to this proposition) shows that L is mildly normal. Consequently, a ∈ Coz L, and therefore L is an Oz-locale. (2) ⇔ (3): This is proved in [17, p. 790]. (1) ⇒ (4): If L is an Oz-locale, then any regular a ∈ L is a cozero element, and so the second part of (4) follows from the normality of Coz L as a σ-frame. For the first part, take countably many an ∈ Coz L such that each an ≺≺ a and a = an . By Corollary 3 in [6], for each n we can find cn ∈ Coz L such that ∞ an ∧ cn = 0 and cn ∨ a = 1. We show that c(a) = o(cn ). Since a ∨ cn = 1, we have c(a) ⊆ o(cn ), which implies c(a) ⊆
∞
n=1
o(cn ). Since cn ≤
n=1
a∗n ,
we have
o(cn ) ⊆ o(a∗n ) ⊆ o(a∗n ) = c(a∗∗ n ), as a consequence of which c(a) ⊆
∞
o(cn ) ⊆
n=1
Therefore c(a) =
∞
∞
a∗∗ = c(a). n
c(a∗∗ n )=c
n=1
o(cn ).
n=1
(4) ⇒ (2): Let a be a regular element in L, and find countably many cozero elements cn such that ∞ c(a) = o(cn ). Then, for every n, c(a) ⊆ o(cn ), which implies a ∨ cn = 1. By the second part of the present n=1
hypothesis, c(a) is completely separated from c(cn ), hence we can find a cozero element dn such that dn ≺ a and dn ∨ cn = 1. Since Coz L is a normal σ-frame, we can thus find, for each n, un ≺ cn such that a ∨ un = 1. This of course implies u∗n ≺ a. Now,
c
∞ ∞ ∞ u∗n = o(un ) ⊆ c(u∗n ) = o(cn ) = c(a), n=1
n=1
n=1
which implies a ≤ u∗n , hence a = u∗n . Therefore a is δ-regular. (1) ⇒ (5): Suppose L is an Oz-locale, and let Q be an annihilator ideal in RL. Then there is an a ∈ L such that Q = M a∗ . Since L is an Oz-locale, a∗ ∈ Coz L, hence Q is a principal z-ideal, by Lemma 3.3, and therefore RL is an Oz-ring. (5) ⇒ (6): This is immediate. (6) ⇒ (1): We apply [4, Proposition 2.2(e)]. Let a ∧ b = 0 in L. Then a∗∗ ∧ b∗∗ = 0, and so M a∗∗ and M b∗∗ are annihilator ideals in RL with M a∗∗ ∩ M b∗∗ = {0}. By (6), there exist α, β ∈ RL such that M (α) ∩ M (β) = {0}, M a∗∗ ⊆ M (α) and M b∗∗ ⊆ M (β). The equality M (α) ∩ M (β) = {0} implies αβ = 0, whence coz α ∧ coz β = 0. Since a ≤ a∗∗ ≤ coz α and b ≤ b∗∗ ≤ coz β, it follows from [4, Proposition 2.2(e)] that L is an Oz-locale. 2 Remark 3.5. The equivalence (1) ⇔ (3) brings the following to mind. In [15, Proposition 3.5] it is proved that, in normal locales, c(a) is a Gδ -sublocale if and only if a is a cozero element. Thus, a normal locale is an Oz-locale if and only if every regular-closed sublocale is a Gδ -sublocale. If, analogously with spaces, we say a locale L is a Moscow locale in case every regular-closed sublocale is a join of Gδ -sublocales, then the equivalence (1) ⇔ (4) shows that every Oz-locale is a Moscow locale. Of course the converse is known to fail already in spaces (see [1, p. 385], for instance).
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Remark 3.6. The equivalence (1) ⇔ (2) is a variant of [4, Proposition 2.3]. It is observed in [4] that a dense quotient of an Oz-locale is an Oz-locale. Of course if a locale has a dense quotient (even a dense C ∗ -quotient) which is an Oz-locale, it does not follow that the locale is itself an Oz-locale. From the equivalence (1) ⇔ (5) we deduce that if h : L → M is a dense C-quotient map, then L is an Oz-locale if and only if M is an Oz-locale since the ring homomorphism Rh : RL → RM is an isomorphism. We shall now give a characterisation in terms of normal lower semicontinuous functions on a locale. This result extends Lane’s [19, Theorem 1.1 (iii)]. We thank Javier Gutiérrez García and Jorge Picado for showing us that it holds in locales, and also for supplying the proof. It is with their kind permission that we include the result here. We need some background, but we shall be terse about it and refer to the articles [16] and [17] for more details. Let S(L) denote the locale S(L)opp and L(R) be the locale of reals. The ring of arbitrary real functions on L, denoted F(L), is defined in [14] to be the ring R(S(L)). The subring of F(L) consisting of bounded functions is denoted by F∗ (L). In [17] the authors define what it means for an f ∈ F(L) to be normal lower semicontinuous. The set of bounded normal lower semicontinuous functions on L is denoted by NLSC∗ (L). In [16, Example 4.6], the authors define the characteristic function, χS , for each complemented S ∈ S(L) in the following manner. Recall that scales can be used to define localic real functions. The function σ : Q → S(L) defined by ⎧ ⎪ ⎪ ⎨1S(L) σ(p) = S ∗ ⎪ ⎪ ⎩0 S(L)
if p < 0 if 0 ≤ p < 1 if p ≥ 1
is a scale on S(L), and χS ∈ F∗ (L) is the real function it defines. Explicitly, χS maps the generators of the form (p, −) as follows: ⎧ ⎪ ⎪ ⎨1S(L) χS (p, −) = S ∗ ⎪ ⎪ ⎩0 S(L)
if p < 0 if 0 ≤ p < 1 if p ≥ 1.
It is observed in [16, Remark 4.7] that, for any a ∈ L, χo(a) ∈ NLSC∗ (L) if and only if a = a∗∗ . Proposition 3.7 (Gutiérrez García and Picado). A completely regular locale L is an Oz-locale iff for any f ∈ NLSC∗ (L), f (p, −) is a zero-set sublocale for every p ∈ Q. Proof. (⇒) Assume L is an Oz-locale. Let f ∈ NLSC∗ (L) and p ∈ Q. Now, Corollary 3.5 in [17] implies that f (p, −) = r>p f (r, −)◦ . Observe that this is a closed sublocale since the join is contemplated in S(L). Denoting this sublocale by c(ap ), the fact that f is normal lower semicontinuous implies ap = r>p a∗∗ r , as shown in [17, Corollary 3.5]. Since L is an Oz-locale, f (p, −) is a zero-set sublocale since r>p a∗∗ is then a r join of countably many cozero elements, so that it is itself a cozero element. (⇐) Assume the condition holds, and let a ∈ L be regular. By the remark above cited from [16], we have that χo(a) ∈ NLSC∗ (L), and so, by the current hypothesis, the closed sublocale c(a) = χo(a) (0, −) is a zero-set sublocale, which means that a ∈ Coz L. Therefore L is an Oz-locale. 2 To close this section, let us recall from [4] that a locale L is called a weak Oz-locale if a∗ ∈ Coz L for every a ∈ Coz L. Two characterisations of these locales are given in [4] including one that says L is a weak
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Oz-locale if and only if for any a ∈ L and c ∈ Coz L such that a ∧ c = 0, there exists d ∈ Coz L such that a ≤ d and c ∧ d = 0 [4, Proposition 5.3]. Here we give some ring-theoretic characterisations. Definition 3.8. We say a ring A is a weak Oz-ring if every element annihilator of A is a principal z-ideal; and we say A is a weak splitting ring if for any element annihilator I and any principal z-ideal U with I ∩ U = 0, there exists a principal z-ideal V such that U ∩ V = 0 and I ⊆ V . A proof along the lines of the equivalences of (1), (5) and (6) in Proposition 3.4 above establishes the following. Proposition 3.9. The following are equivalent for a completely regular locale L. (1) L is a weak Oz-locale. (2) RL is a weak Oz-ring. (3) RL is a weak splitting ring. 4. When βL is an Oz-locale In this section we shall give some necessary and sufficient conditions for βL to be an Oz-locale under some (necessary) assumption on L. The assumption is the following weakening of normality. In his study of countable paracompactness and weak normality conditions, Mack [20] introduces what he calls weakly δ-normally separated spaces. These are spaces in which every regular-closed set is completely separated from every zero-set disjoint from it. Locales with these features were considered in [13] and were named “weakly δ-normally separated locales”. This is quite a mouthful, so here we shall abbreviate and say such locales are wδ-normal. Thus, paraphrasing [13, Lemma 3.14], we have that L is wδ-normal iff for every regular a ∈ L and c ∈ Coz L with a ∨ c = 1, there exists d ∈ Coz L such that d ≤ a and d ∨ c = 1. By the normality of the σ-frame Coz L, if L is wδ-normal and a ∨ c = 1 for some regular a ∈ L and c ∈ Coz L, then there exists b ∈ Coz L such that b ≺≺ a and b ∨ c = 1. One upshot of this is that we then have rL (a) ∨ rL (c) = 1βL because rL preserves finite joins of cozero elements. Trivially, every Oz-locale is wδ-normal. Lemma 4.1. Let L be a completely regular wδ-normal locale. For any regular a ∈ L, rL (a) ∈ Coz(βL) iff c(a) has a countable cozero neighbourhood base for its cozero neighbourhoods. Proof. (⇒) Assume rL (a) ∈ Coz(βL). By [6, Corollary 4], rL (a) is a Lindelöf element in βL. Since rL (a) =
{rL (t) | t ≺≺ a},
we can find an increasing sequence (tn ) of cozero elements of L such that tn ≺≺ a for each n, and rL (a) =
{rL (tn ) | n ∈ N}.
Since rL (tn ) ≺≺ rL (a) for each n, we can find In ∈ Coz(βL) such that rL (tn ) ∧ In = 0βL
and
In ∨ rL (a) = 1βL .
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Put cn = In , and observe that each cn is a cozero element of L with a ∨ cn = 1. That is, each o(cn ) is a cozero neighbourhood of c(a). Now let o(d), with d ∈ Coz L, be any cozero neighbourhood of c(a). Then a ∨ d = 1, hence, as remarked just prior to the statement of the lemma, rL (a) ∨ rL (d) = 1βL , that is, rL (tn ) ∨ rL (d) = 1βL . n
By compactness of βL, and because the sequence (tn ) is increasing, there is an n such that rL (tn ) ∨ rL (d) = 1βL . Since In ∧ rL (tn ) = 0βL , we have In ≤ rL (d), whence cn ≤ d. Therefore o(cn ) ⊆ o(d), which establishes the implication. (⇐) Assume the stated condition holds, and find countably many cn ∈ Coz L such that {o(cn ) | n ∈ N} is a base for the cozero neighbourhoods of c(a). Then a ∨ cn = 1 for every n. Since L is wδ-normal and a is regular, we can find, for each n, an element un ∈ Coz L such that un ≺≺ a and un ∨ cn = 1. We claim that rL (a) =
rL (un ).
n
The containment
rL (un ) ⊆ rL (a) is immediate. To show the other containment, let t ≺≺ a, and pick
n
w ∈ Coz L with t ≺≺ w ≺≺ a. Next, take s ∈ Coz L such that w ∧ s = 0 and s ∨ a = 1. Then o(s) is a cozero neighbourhood of c(a), hence, by the present hypothesis, there is an m such that cm ≤ s. Then w ∧ cm = 0, and hence w ≺≺ um (with cm as a separating element), which implies t ≺≺ um , so that t ∈ rL (um ). Thus rL (a) ⊆
n
rL (un ) ⊆
rL (un ),
n
and hence equality. Since rL (un ) ≺≺ rL (a) for each n, it follows that rL (a) ∈ Coz(βL).
2
Let us observe the following. If h : M → L is a dense frame homomorphism, then h∗ h(a) = a for every regular a ∈ M . Indeed, the equality h h∗ h(a) ∧ a∗ = 0 implies h∗ h(a) ∧ a∗ = 0 by density of h, whence h∗ h(a) ≤ a∗∗ = a, and hence the claimed statement. Proposition 4.2. The following are equivalent for a wδ-normal locale L. (1) βL is an Oz-locale. (2) Every regular-closed sublocale of L has a countable cozero neighbourhood base for its cozero neighbourhoods. Proof. (1) ⇒ (2): Let a ∈ L be regular. Then rL (a) is a regular element in βL, and so rL (a) ∈ Coz(βL) since βL is an Oz-locale. Therefore, by Lemma 4.1, c(a) has a countable cozero neighbourhood base for its cozero neighbourhoods. (2) ⇒ (1): Let I be a regular element of βL, and write a = I. Then a is a regular element of L, and I = rL (a) by what we observed above. By the current hypothesis, c(a) has a countable cozero neighbourhood base for its cozero neighbourhoods, which then makes rL (a) a cozero element in βL by Lemma 4.1. Therefore βL is an Oz-locale. 2 For our next characterisation we need some terminology, notation, and three lemmas. First, following [9], we say a sublocale S of a locale L is peripheral if it is of the form c(a ∨ a∗ ), for some regular a ∈ L. Next, for any a ∈ L we write κa : L → ↑a for the frame homomorphism x → a ∨ x. We present the first lemma in frame language. Let us therefore recall complete separation of quotients β α in frame terms. Let A ←− L −→ B be quotients of L, which is to say α and β are surjective frame
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homomorphisms. The quotients A and B are completely separated if there exist c, d ∈ Coz L with c ∨ d = 1 such that α(c) = 0A and β(d) = 0B . Note that for the homomorphism κa : L → ↑a, κ(c) = 0↑a if and only if c ≤ a. Recall that if u ∨ v = 1 in Coz L, then there exists w ∈ Coz L such that w ≺≺ v and u ∨ w = 1. h
Lemma 4.3. Let L −→ M be a quotient of L. Suppose that there is a sequence (an) in L such that, for every c ∈ Coz L with h(c) = 1, there is an index m with am ≤ c. Suppose further that, for each n, the quotients κa
h
n ↑an ←− L −→ M are completely separated. Then rL h∗ (0) ∈ Coz(βL).
κa
h
un ∨ vn = 1,
and
n Proof. By complete separation of the quotients ↑an ←− L −→ M , there exist, for each n, cozero elements of un and vn of L such that
un ≤ an ,
h(vn ) = 0.
Then vn ≤ h∗ (0). Now take wn ∈ Coz L such that wn ≺≺ vn and un ∨ wn = 1. We claim that rL h∗ (0) = The containment
{rL (wn ) | n ∈ N}.
rL (wn ) ⊆ rL h∗ (0) is immediate because wn ≤ h∗ (0) for every n. Now let t ≺≺ h∗ (0), n
and pick s ∈ Coz L such that t ∧ s = 0 and s ∨ h∗ (0) = 1. Then h(s) = 1, and so, by hypothesis, there is an index m such that am ≤ s. Now, t ∧ u m ≤ t ∧ am ≤ t ∧ s = 0
and
um ∨ wm = 1,
from which we can deduce that t ≺≺ wm because the separating element um is a cozero element in L. Therefore t ∈ rL (wn ), so that rL h∗ (0) ⊆ rL (wn ), and hence equality. Since rL (wn ) ≺≺ rL h∗ (0) for n
n
each n, it follows that rL h∗ (0) is a cozero element in βL. 2 Lemma 4.4. Let a and b be regular elements in a locale L. If a ∨ a∗ ∨ b = 1, then a∗ ∨ b is regular. Proof. Since a ∨ (a∗ ∨ b) = 1, (a∗ ∨ b)∗∗ = (a∗ ∨ b)∗∗ ∧ a ∨ (a∗ ∨ b)∗∗ ∧ (a∗ ∨ b) = (a∗ ∨ b)∗∗ ∧ a∗∗ ∨ (a∗ ∨ b) ∗∗ = (a∗ ∨ b) ∧ a ∨ (a∗ ∨ b) = (a ∧ b)∗∗ ∨ (a∗ ∨ b) = (a ∧ b) ∨ (a∗ ∨ b)
since a and b are regular
= (a∗ ∨ b), which proves the result. 2 Finally, we shall also need the following lemma which appears as Proposition 3.3.6 in [23]. We shall give only an outline of the proof. Lemma 4.5. Let L be a mildly normal frame. If a ∨ b = 1 where a and b are regular elements of L, then the κb κa quotients ↑a ←− L −→ ↑b are completely separated.
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Proof. One shows first that if s and t are regular elements in L with s ≺ t, then for some regular r ∈ L we have s ≺ r ≺ t. The proof mimics the one that shows that the rather below relation interpolates in normal frames. Here, the rather below relation between regular elements interpolates through a regular element. Thus, by Countable Dependent Choice, s ≺ t implies s ≺≺ t. Then the proof of [2, Proposition 8.3.1] is copied verbatim (except that one restricts to regular elements) to show that there exist c, d ∈ Coz L such that c ≤ a, d ≤ b and c ∨ d = 1. This establishes the complete separation of the quotients. 2 An immediate consequence of this is that if L is mildly normal and a, b ∈ L are regular, then rL (a) ∨ rL (b) = 1βL . Proposition 4.6. The following are equivalent for a mildly normal completely regular locale L. (1) βL is an Oz-locale. (2) Every regular-closed sublocale of L has a countable regular-open neighbourhood base for its regular-open neighbourhoods. (3) Every peripheral sublocale of L has a countable regular-open neighbourhood base for its regular-open neighbourhoods. Proof. (1) ⇒ (2): Let a be a regular element in L. Then rL (a) is regular in βL, and is therefore a cozero element because βL is an Oz-locale. Thus, there are elements In in βL such that In ≺≺ In+1 for every n, and rL (a) = In . For each n choose a regular Jn in βL such that n
In ∧ Jn = 0βL
and
rL (a) ∨ Jn = 1βL .
For each n let bn = Jn , and note that each bn is a regular element of L. Also, applying the map jL : βL → L to the equality rL (a) ∨ Jn = 1βL , we see that a ∨ bn = 1 for each n, so that each o(bn ) is a neighbourhood of c(a). Let x be a regular element of L with a ∨ x = 1. Since L is mildly normal, rL (a) ∨ rL (x) = 1βL . That is, rL (x) ∨ In = 1βL , which, by the compactness of βL, and the fact that the sequence (In ) is increasing, n
implies that there is an index k such that rL (x) ∨ Ik = 1βL . Since Ik ∧ Jk = 0βL , it follows that Jk ≤ rL (x), and hence bk = Jk ≤ rL (x) = x, so that o(bk ) ⊆ o(x). Therefore the set {o(bn ) | n ∈ N} is a countable regular-open neighbourhood base for the regular-open neighbourhoods of c(a). (2) ⇒ (1): Let I be a regular element of βL. Then I = rL (a) for the regular element a = I of L. By the current hypothesis, there are regular elements un in L such that the collection {o(un ) | n ∈ N} is a neighbourhood base for the regular-open neighbourhoods of c(a). Observe that, for each n, the quotients κun κa ↑un ←− L −→ ↑a are completely separated because a ∨ un = 1 and L is mildly normal. Furthermore, if κa (u) = 1 for some u ∈ L, then a ∨ u = 1, and hence un ≤ u for some n. Therefore, by Lemma 4.3, rL (κa )∗ (0↑a ) ∈ Coz(βL). But rL (κa )∗ (0↑a ) = rL (a), so we are done. (2) ⇔ (3): Assume (2) and consider any peripheral sublocale c(a ∨ a∗ ) of L, with a regular. Let {o(un ) | n ∈ N} and {o(vn ) | n ∈ N} be regular-open neighbourhood bases for the regular-open neighbourhoods of c(a) and c(a∗ ), respectively, which are guaranteed by (2). Let o(x) be a regular-open neighbourhood of c(a ∨a∗ ). By Lemma 4.4, a ∨x and a∗ ∨x are regular elements, hence o(a∗ ∨x) is a regular-open neighbourhood of c(a), and o(a ∨ x) is a regular-open neighbourhood of c(a∗ ). Take k, ∈ N such that o(uk ) ⊆ o(a∗ ∨ x) and o(v ) ⊆ o(a ∨ x). A simple calculation shows that o(uk ∧ v ) ⊆ o(x). This shows that the collection {o(um ∧ vn ) | m, n ∈ N} is a countable regular-open neighbourhood base for regular-open neighbourhoods of c(a ∨ a∗ ). Conversely, assume (3) holds and let c(a) be a regular-closed sublocale of L. Let {o(wn ) | n ∈ N} be a regular-open neighbourhood base for the regular-open neighbourhoods of c(a ∨ a∗ ). By Lemma 4.4, each o(a∗ ∨ wn ) is a regular-open neighbourhood of c(a). Let o(x) be a regular-open neighbourhood of c(a). Then
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o(x) is also a regular-open neighbourhood of c(a ∨ a∗ ), and so there exists m ∈ N such that o(wm ) ⊆ o(x). Since a ∨ x = 1 (as c(a) ⊆ o(x)), a∗ ≤ x, which then implies a∗ ∨ wn ≤ x, whence o(a∗ ∨ wm ) ⊆ o(x). We then deduce that {o(a∗ ∨ wn ) | n ∈ N} is a countable regular-open neighbourhood base for regular-open neighbourhoods of c(a). 2 Borrowing terminology from spaces [25], we say a locale L is almost normal if every regular-closed sublocale of L is completely separated from every closed sublocale disjoint from it. It is easy to show that if L is almost normal and a ∨ b = 1 with a regular, then there is a c ≺ b such that a ∨ c = 1. We therefore have the following corollary. Corollary 4.7. The following are equivalent for an almost normal locale L. (1) βL is an Oz-locale. (2) Every regular-closed sublocale of L has a countable basis for its neighbourhoods. Let us recall that λL is always a normal locale and β(λL) ∼ = βL. We can thus put together the various characterisations above of when βL is an Oz-locale into a single corollary stated in terms of λL as follows. Corollary 4.8. The following are equivalent for any completely regular locale L. (1) βL is an Oz-locale. (2) Every regular-closed sublocale of λL has a countable cozero neighbourhood base for its cozero neighbourhoods. (3) Every regular-closed sublocale of λL has a countable regular-open neighbourhood base for its regular-open neighbourhoods. (4) Every regular-closed sublocale of λL has a countable basis for its neighbourhoods. (5) Every peripheral sublocale of λL has a countable regular-open neighbourhood base for its regular-open neighbourhoods. In closing, let us consider when βL is a weak Oz-locale. Observe that if L is a weak Oz-locale, then for any a ∈ Coz L, rL (a∗ ) ∨ rL (c) = 1βL whenever a and c are cozero elements with a∗ ∨ c = 1. In the proof of Lemma 4.1 the fact that L is a wδ-normal locale was used to deduce that rL (a) ∨ rL (c) = 1βL whenever a ∨ c = 1 for regular a and c ∈ Coz L. Hence, the same proof yields the following lemma. A support of L is a closed sublocale of the form c(a∗ ), for a ∈ Coz L. This term is of course borrowed from classical topology and agrees with its topological namesake because c(a∗ ) = o(a). Lemma 4.9. Let L be a weak Oz-locale. For any a ∈ Coz L, rL (a∗ ) ∈ Coz(βL) iff c(a∗ ) has a countable cozero neighbourhood base for its cozero neighbourhoods. Proposition 4.10. The following are equivalent for a weak Oz-locale L. (1) βL is a weak Oz-locale. (2) Every support of L has a countable cozero neighbourhood base for its cozero neighbourhoods. Proof. Assume βL is a weak Oz-locale and consider any support c(a∗ ) of L with a ∈ Coz L. Since jL : βL → L is coz-onto, there exists I ∈ Coz(βL) such that I = a. Then rL (a∗ ) = I ∗ ∈ Coz(βL) since βL is a weak Oz-locale. Therefore c(a∗ ) has the stated property by the foregoing lemma.
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Conversely, suppose the stated condition holds. Let I ∈ Coz(βL) and put a = I. Then a ∈ Coz L, and hence a∗ ∈ Coz L since L is a weak Oz-locale. Now, I ∗ = rL (a∗ ), whence I ∗ ∈ Coz(βL) by the lemma. Therefore βL is a weak Oz-locale. 2 There are also characterisations for when βL is a weak Oz-locale analogous to those in Proposition 4.6 under the hypothesis that L is a weak Oz-locale. Let us say a sublocale of L is a co-support if it is of the form o(c∗ ), with c ∈ Coz L; and that it is coz-peripheral if it is of the form c(c∗ ∨ c∗∗ ), for some c ∈ Coz L. Observe that if L is a weak Oz-locale, then c∗∗ ∈ Coz L for every c ∈ Coz L, hence, by Lemma 4.4, for any d ∈ Coz L with c∗ ∨ c∗∗ ∨ d∗ = 1, c∗∗ ∨ d∗ is a pseudocomplement of some cozero element of L. Using this observation (as a “weak Oz” analogue of Lemma 4.4), we can prove along similar lines to Proposition 4.6 the following. Proposition 4.11. The following are equivalent for a weak Oz-locale L. (1) βL is a weak Oz-locale. (2) Every support of L has a countable coz-support neighbourhood base for its coz-support neighbourhoods. (3) Every coz-peripheral sublocale of L has a countable coz-support neighbourhood base for its coz-support neighbourhoods. Acknowledgements Thanks are due to the referee for detailed comments and helpful suggestions which have improved this paper. References [1] A.V. Arhangel’skii, Moscow spaces and topological groups, Topol. Proc. 25 (2000) 383–416. [2] R.N. Ball, J. Walters-Wayland, C- and C ∗ -quotients in pointfree topology, Diss. Math. (Rozprawy Mat.) 412 (2002), 62 pp. [3] B. Banaschewski, The real numbers in pointfree topology, Textos de Matemática Série B, vol. 12, Departamento de Matemática da Universidade de Coimbra, 1997. [4] B. Banaschewski, T. Dube, C. Gilmour, J. Walters-Wayland, Oz in pointfree topology, Quaest. Math. 32 (2009) 215–227. [5] B. Banaschewski, C. Gilmour, Stone–Čech compactification and dimension theory for regular σ-frames, J. Lond. Math. Soc. 39 (1985) 1–8. [6] B. Banaschewski, C. Gilmour, Pseudocompactness and the cozero part of a frame, Comment. Math. Univ. Carol. 37 (1996) 577–587. [7] B. Banaschewski, C. Gilmour, Oz revisited, in: H. Herrlich, H.-E. Porst (Eds.), Proceedings of the Conference on Categorical Methods in Algebra and Topology, in: Math. Artbeitspapiere, vol. 54, Universität Bremen, 2000, pp. 19–23. [8] R.L. Blair, Spaces in which special sets are z-embedded, Can. J. Math. 28 (4) (1976) 673–690. [9] R.L. Blair, M.A. Swardson, Spaces with an Oz Stone–Čech compactification, Topol. Appl. 36 (1990) 73–92. [10] T. Dube, Some ring-theoretic properties of almost P -frames, Algebra Univers. 60 (2009) 145–162. [11] T. Dube, Contracting the socle in rings of continuous functions, Rend. Semin. Mat. Univ. Padova 123 (2010) 37–53. [12] T. Dube, O. Ighedo, On z-ideals of pointfree function rings, Bull. Iran. Math. Soc. 40 (2014) 655–673. [13] T. Dube, M. Matlabyana, Concerning variants of C-embedding in pointfree topology, Topol. Appl. 158 (2011) 2307–2321. [14] J. Gutiérrez García, T. Kubiak, J. Picado, Localic real functions: a general setting, J. Pure Appl. Algebra 213 (2009) 1064–1074. [15] J. Gutiérrez García, T. Kubiak, J. Picado, Pointfree forms of Dowker’s and Michael’s insertion theorems, J. Pure Appl. Algebra 213 (2009) 98–108. [16] J. Gutiérrez García, I. Mozo Carolo, J. Picado, Normal semicontinuity and the order completion of pointfree function rings (Preprint). [17] J. Gutiérrez García, J. Picado, On the parallel between normality and extremal disconnectedness, J. Pure Appl. Algebra 218 (2014) 784–803. [18] P.B. Johnson, κ-Lindelöf locales and their spatial parts, Cah. Topol. Géom. Différ. Catég. 32 (1991) 297–313. [19] E.P. Lane, PM-normality and the insertion of a continuous function, Pac. J. Math. 82 (1979) 155–162. [20] J. Mack, Countable paracompactness and weak normality properties, Trans. Am. Math. Soc. 148 (1970) 265–272. [21] J. Madden, J. Vermeer, Lindelöf locales and realcompactness, Math. Proc. Camb. Philos. Soc. 99 (1986) 473–480. [22] G. Mason, z-Ideals and prime ideals, J. Algebra 26 (1973) 280–297.
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