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The Vietoris uniformity for locales
Journal Pre-proof The Vietoris uniformity for locales
Dharmanand Baboolal, Paranjothi Pillay
PII:
S0166-8641(19)30384-0
DOI:
https://doi.org/10.1016/j.topol.2019.106979
Reference:
TOPOL 106979
To appear in:
Topology and its Applications
Received date:
9 December 2018
Revised date:
29 March 2019
Accepted date:
1 April 2019
Please cite this article as: D. Baboolal, P. Pillay, The Vietoris uniformity for locales, Topol. Appl. (2019), 106979, doi: https://doi.org/10.1016/j.topol.2019.106979.
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THE VIETORIS UNIFORMITY FOR LOCALES DHARMANAND BABOOLAL AND PARANJOTHI PILLAY Dedicated to Aleˇs Pultr on his 80th birthday
Abstract. The purpose of this paper is to define the Vietoris uniformity on the Vietoris locale of a uniform locale A. This is the pointfree counterpart to the classical construction of the hyperspace uniformity on the hyperspace of non-empty compact subsets of a uniform space.
1. Introduction The purpose of this paper is to define the Vietoris uniformity on the Vietoris locale of a uniform locale A. The Vietoris locale V (A) of an arbitrary locale A was defined by Johnstone [6], and it is the analogue of the hyperspace of all compact subsets, including the empty set, of an arbitrary topological space (see e.g. [3], [7]). However, we will focus specifically on the pointfree analogue of the hyperspace of all non-empty compact subsets of a space. This amounts to working with a specific closed sublocale of V (A), what Johnstone refers to as V + (A) in his paper. We do this because in the classical theory of hyperspaces one works only with the non-empty subsets of a space, and we wanted to capture this theory in the pointfree context. To simplify the notation we will rename V + (A) as H(A) throughout the paper. One of the things considered by Johnstone in his paper was to see which properties possessed by A would be inherited by V (A). He showed this is true for the properties of regularity, complete regularity and 0-dimensionality, amongst others. Our paper can be seen as an extension of this idea in the sense that if we start with a uniformity on A we show that this uniformity can be extended to a uniformity on H(A), i.e. A can be uniformly embedded in H(A), and in fact as a closed sublocale of H(A). We also show that this construction makes H functorial. We also show that if H(A) is complete then so is A, but we leave as an open question whether the converse is also true. 2. Preliminaries A frame A is a complete lattice which satisfies the infinite distributive law: x ∧ S = {x ∧ s|s ∈ S} for all x ∈ A, S ⊆ A. The top element of A is denoted by 1 and the bottom by 0. A frame homomorphism is a map f : A → B between frames that preserves finitary meets (including the top 1) and arbitrary joins (including the bottom 0). We thus have the category of frames and frame homomorphisms, which we denote by Frm. A frame map f : A → B is said to be dense if x = 0 whenever f (x) = 0. Recall that every frame homomorphism f : A → B has a right adjoint f∗ : B → A characterized by the condition f (x) ≤ y ⇔ x ≤ f∗ (y). Date: 30 March 2019. 1991 Mathematics Subject Classification. 06D22, 54D35, 54E15. Key words and phrases. frame, locale, Vietoris locale, uniformity, completion. A grant from the National Research Foundation (S.A), Grant No. 113840, is gratefully acknowledged. 1
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DHARMANAND BABOOLAL AND PARANJOTHI PILLAY
If f is onto then f∗ is given by the formula f∗ (y) = {x ∈ A|f (x) = y}. It follows that for such f the composite map f f∗ is the identity map. For every pair of elements a, b in a frame we have the element a → b given by the Heyting operation → characterised by the condition x ≤ a → b iff x ∧ a ≤ b. The category Loc is the category with objects frames (or locales) and with morphisms f : A → B where f is the right adjoint of some frame map g : B → A. Sublocales of a locale A are the regular subobjects of A in Loc, or in Frm the quotients of A. Sublocales are those subsets S of A having the characteristic properties: (1) If M ⊆ S then M ∈ S. (2) If a ∈ A, s ∈ S then a → s ∈ S. The collection of sublocales S(A) of a locale A ordered by inclusion form a co-frame, that is, Ti = (S ∨ Ti ) S∨ i∈I
i∈I
for sublocales S and Ti . Here, the infimum of sublocales is just theset-theoretic intersection, and the supremum of a collection of sublocales Ti is the sublocale { M |M ⊆ Ti }. An open sublocale is one of the form o(a) = {a → x|x ∈ A} = {x ∈ A|x = a → x}, and a closed sublocale is one of the form c(a) =↑ a. The sublocales o(a) and c(a) are complements of each other in the sublocale lattice. We also have o(
o(0) =O i ai ) = i o(ai )
↑(
A o(1) = i ai ) = i ↑ ai
o(a ∧ b) = o(a) ∩ o(b) ↑ (a ∧ b) =↑ a ∨ ↑ b
where O = {1} is the trivial sublocale. A frame homomorphism f : A → B is said to be closed if f∗ (b ∨ f (a)) = f∗ (b) ∨ a for all a ∈ A and all b ∈ B. This is equivalent to saying that the localic map f∗ maps closed sublocales of B to closed sublocales of A. For elements a, b ∈ A, we say that a is rather below b, written a ≺ b, if there exists an element c ∈ A such that a ∧ c = 0 and c ∨ b = 1. This is equivalent to the condition that a∗ ∨ b = 1, where a∗ is the pseudocomplement a → 0 of a, i.e. the largest element in A whose meet with a is 0. A frame A is called regular if for each a ∈ A, a = {x ∈ A|x ≺ a}. A frame A is called completely regular if for each a ∈ A, a = {x ∈ A|x ≺≺ a} where x ≺≺ a means that there exists a doubly indexed sequence of elements in A, (xnk )n=0,1,...;k=0,1,...,2n such that x = xn0 , xnk ≺ xnk+1 , xn2n = a, xnk = xn+1 2k for all n = 0, 1, ... and k = 0, 1, ..., 2n . A frame A is said to be 0-dimensional if it has a basis consisting of complemented elements, i.e. elements a such that a ∨ a∗ = 1. A cover of a frame A is a subset C of A such that C = 1. A cover C is said to refine a cover D (written as C ≤ D) if for each c ∈ C there exists d ∈ D such that c ≤ d. A cover ∗ C is said to star-refine a cover D (written as C ≤ D) if the cover {Cx|x ∈ C} ≤ D, where Cx = {c ∈ C|c ∧ x = 0}. A uniform frame (A, μ) is a frame A together with a specified collection μ of covers, called uniform covers, satisfying the following properties: (i) If C ∈ μ and C ≤ D, then D ∈ μ. (ii) If C, D ∈ μ, then C ∧ D ∈ μ, where C ∧ D = {c ∧ d|c ∈ C, d ∈ D}. (iii) If C ∈ μ then there exists D ∈μ such that D ≤∗ C. (iv) For each a ∈ A we have a = x(x a), where x a means that there exists some C ∈ μ such that Cx ≤ a. A uniformity basis μ0 for a uniformity μ on A is a subcollection of μ such that for each C ∈ μ there exists a D ∈ μ0 such that D ≤ C. A uniform frame is said to be of countable type if the uniform covers have a countable basis.
THE VIETORIS UNIFORMITY FOR LOCALES
3
A frame map f : (A, μ) → (B, ν) between uniform frames is called uniform if f (C) ∈ ν whenever C ∈ μ. If, furthermore, f is onto and for each D ∈ ν there exists C ∈ μ such that f (C) ≤ D, then we say that f is a surjection. We also say in this situation that B is uniformly embedded in A. The uniform frame (B, ν) is said to be complete if whenever (A, μ) is a uniform frame such that f : (A, μ) → (B, ν) is a dense surjection, then f is an isomorphism. The category of uniform frames and uniform frame maps will be denoted by UniFrm. For general background on frames the reader is referred to the books [5] and [9]. 3. The Vietoris Locale To make this paper self-contained we recall Johnstone’s construction of the Vietoris locale and some of its basic facts (see [6]). Let A be a locale. For each a ∈ A, let t(a) and m(a) be abstract symbols. Then V (A) is the frame freely generated by these symbols subject to the following relations: t(a ∧ b) = t(a) ∧ t(b) for all a, b ∈ A, and t(1) = 1. t( S) = t(s)(s ∈ S) for all updirected S ⊆ A. m( S) = m(s)(s ∈ S) for all S ⊆ A, and m(0) = 0.
(iii)
m(a ∧ b) ≥ t(a) ∧ m(b) for all a, b ∈ A.
(iv)
t(a ∨ b) ≤ t(a) ∨ m(b) for all a, b ∈ A.
(v)
(i) (ii)
As Johnstone remarks in his paper one can think informally of V (A) as the space of all compact subspaces of A, of t(a) as the set of those compact subspaces contained in a, and of m(a) as the set of those compact subspaces that meet a. Let X = {m(a), t(a)|a ∈ A}. The above relations give the construction of V (A) diagrammatically as shown below: X
FX f
f¯
ν
F X/Θ = V (A) ϕ
A Here f is the map t(a) −→ a, and m(a) −→ a, F X is the free frame on X, X → F X is the insertion of generators, and f¯ is the unique frame homomorphism making the left triangle in the diagram commute. It is easily verified that the relation R on F X determined by the relations (i)-(v) is such that R ⊆ ker f¯. Thus if Θ is the congruence on F X generated by R, and ν the corresponding quotient map, we obtain a unique frame homomorphism ϕ : F X/Θ → A making the second triangle in the diagram commute. The Vietoris locale V (A) is defined to be the frame F X/Θ. (See [9, IV.2] for a discussion on free frame constructions). 3.1. Some properties of V (A). If w ∈ F X, then ν(w) ∈ V (A). We will write ν(w) as w, i.e. we will suppress the quotient map ν. Bearing this in mind, we then see that ϕ(t(a)) = a and ϕ(m(a)) = a. Hence: (a) ϕ : V (A) → A is an onto frame homomorphism. Thus, A is a sublocale of V (A). If A is regular, then V (A) is regular and in this case ϕ : V (A) → A is also closed. Thus A is embedded as a closed sublocale of V (A) ([6]). (b) t(0) and m(1) are complements of each other in V (A). To see this, note from the relation (iv) above that t(0) ∧ m(1) ≤ m(0 ∧ 1) = m(0) = 0. Thus t(0) ∧ m(1) = 0. Also 1 = t(1) = t(0 ∨ 1) ≤ t(0) ∨ m(1), hence t(0) ∨ m(1) = 1. Of course m(0) and t(1) are also complements of each other in V (A) ([6]).
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DHARMANAND BABOOLAL AND PARANJOTHI PILLAY
(c) Every element x of V (A) is a join of elements of the type t(a1 ) ∧ t(a2 ) ∧ . . . ∧ t(am ) ∧ m(b1 ) ∧ m(b2 ) ∧ . . . ∧ m(bn ). Since t preserves finite meet, a basic generator has the form t(a) ∧ m(b1 ) ∧ m(b2 ) ∧ . . . ∧ m(bn ). But from relation (iv) of the definition of V (A) we have t(a) ∧ m(b1 ) ∧ m(b2 ) ∧ . . . ∧ m(bn ) = t(a) ∧ m(a ∧ b1 ) ∧ m(a ∧ b2 ) ∧ . . . ∧ m(a ∧ bn ). Thus the basic generators have the form t(a) ∧ m(b1 ) ∧ m(b2 ) ∧ . . . ∧ m(bn ) where bi ≤ a for each i ([6]). (d) From (b) above we see that V (A) can be written as the disjoint join of the two closed sublocales ↑ m(1) and ↑ t(0), what is referred to by Johnstone as V0 (A) and V + (A) respectively. Of course these sublocales are also open, hence they are clopen ([6]). (e) If one adjoins the relation t(0) = 1 to those relations (i)-(v) that define V (A), then one gets the sublocale V0 (A). This is because t(0) = 1 ⇔ m(1) = 0, since t(0) and m(1) are complementary. It is shown in [6] that V0 (A) ∼ = 2, where 2 is the terminal object in Loc. (f) If one adjoins the relation t(0) = 0 to those relations (i)-(v) that define V (A), then one gets the sublocale V + (A) of V (A). The sublocale V + (A) corresponds to the hyperspace of all non-empty compact subsets in the setting of spaces. Since V + (A) would be our primary interest of study from now on, it may be better to change notation and refer to V + (A) as H(A). Thus H(A) is the Vietoris locale of all “non-empty compact subspaces” of A. The relations (i)-(v) as well as (vi) t(0) = 0, then determine H(A), and we can represent this diagrammatically as X
FX f
f¯
ν
F X/Φ = H(A) g
(3.1.1)
A Just as before, g is a frame homomorphism which is onto since g(m(a)) = a and g(t(a)) = a for all a ∈ A. The map g is also closed if A is regular. This is because the map ϕ : V (A) → A was shown to be closed for such A ([6]), and H(A) has one more defining relation than V (A). 3.2. Some properties of H(A). The extra relation t(0) = 0 means that H(A) satisfies some properties not enjoyed by V (A). We list these useful properties which are of crucial importance in the sequel. (a) t(0) = 0 ⇔ t(a) ≤ m(a) for all a ∈ A. To see this note that t(a) = t(0 ∨ a) ≤ t(0) ∨ m(a) = 0 ∨ m(a) = m(a). For the other direction, if t(a) ≤ m(a) for all a, then t(0) ≤ m(0) = 0. Hence t(0) = 0. (b) m(1) = 1. This follows since, as we saw before, t(0) and m(1) are complementary. Proposition 3.1. In H(A), the collection T = {t(a) ∧ m(b1 ) ∧ m(b2 ) ∧ . . . ∧ m(bn )| a = b1 ∨ b2 ∨ . . . ∨ bn , a ∈ A, bi ∈ A, n ∈ N} is a basis for H(A).
THE VIETORIS UNIFORMITY FOR LOCALES
5
Proof. Take a basic generator of H(A), say, t(a) ∧ m(b1 ) ∧ m(b2 ) ∧ . . . ∧ m(bn ) with bi ≤ a for all i. Since t(a) ≤ m(a) in H(A), we have t(a) ∧ m(b1 ) ∧ m(b2 ) ∧ . . . ∧ m(bn ) = t(a) ∧ m(a) ∧ m(b1 ) ∧ m(b2 ) ∧ . . . ∧ m(bn ) with a ∨ b1 ∨ b2 ∨ . . . ∨ bn = a, and the latter is in T .
The following lemma will be useful in the section on the Vietoris uniformity. Lemma 3.2. For elements a1 , a2 , . . . , an , an+1 in A, we have: t(a1 ∨ . . . ∨ an+1 ) ∧ m(a1 ) ∧ . . . ∧ m(an ) ≤ ≤[t(a1 ∨ . . . ∨ an+1 ) ∧ (m(a1 ) ∧ m(a2 ) ∧ . . . m(an+1 ))] ∨ [t(a1 ∨ . . . ∨ an ) ∧ (m(a1 ) ∧ . . . ∧ m(an ))]. Proof. The left hand side is less than or equal to [t(a1 ∨ . . . ∨ an ) ∨ m(an+1 )] ∧ [m(a1 ) ∧ . . . ∧ m(an )] = [t(a1 ∨ . . . ∨ an ) ∧ m(a1 ) ∧ . . . ∧ m(an )] ∨ [m(a1 ) ∧ . . . ∧ m(an+1 )] ≤ t(a1 ∨ . . . ∨ an+1 ) ∧ {[t(a1 ∨ . . . ∨ an ) ∧ m(a1 ) ∧ . . . ∧ m(an )] ∨ [m(a1 ) ∧ . . . ∧ m(an+1 )]} = [t(a1 ∨ . . . ∨ an ) ∧ m(a1 ) ∧ . . . ∧ m(an )] ∨ [t(a1 ∨ . . . ∨ an+1 ) ∧ m(a1 ) ∧ . . . ∧ m(an+1 )].
We note that the two terms in square brackets that appear in the last line in the above proof each have the following property: The join of the arguments of m is the argument of t. We shall refer to these terms as terms of type A. The above lemma allows us to make some computations: t(a1 ∨ a2 ) ∧ m(a1 ) ≤ [t(a1 ∨ a2 ) ∧ m(a1 ) ∧ m(a2 )] ∨ [t(a1 ) ∧ m(a1 )]. Thus the left hand side of the above expression is less than or equal to the join of terms of type A. For a little more involved expression: t(a1 ∨ a2 ∨ a3 ∨ a4 ) ∧ m(a1 ) ≤ [t(a1 ∨ a2 ∨ a3 ∨ a4 ) ∧ m(a1 ) ∧ m(a2 ∨ a3 ∨ a4 )] ∨ [t(a1 ) ∧ m(a1 )] = [t(a1 ∨ a2 ∨ a3 ∨ a4 ) ∧ m(a1 ) ∧ (m(a2 ) ∨ m(a3 ) ∨ m(a4 ))] ∨ [t(a1 ) ∧ m(a1 )] = [t(a1 ∨ a2 ∨ a3 ∨ a4 ) ∧ m(a1 ) ∧ m(a2 )] ∨ . . . ∨ . . . ∨ [t(a1 ) ∧ m(a1 )]. If we consider the expression t(a1 ∨ a2 ∨ a3 ∨ a4 ) ∧ m(a1 ) ∧ m(a2 ), we see that we can use the above lemma to get this expression to be less than or equal to [t(a1 ∨ a2 ∨ a3 ∨ a4 ) ∧ m(a1 ) ∧ m(a2 ) ∧ m(a3 ∨ a4 )] ∨ [t(a1 ∨ a2 ) ∧ m(a1 ) ∧ m(a2 )] and this is equal to [t(a1 ∨ a2 ∨ a3 ∨ a4 ) ∧ m(a1 ) ∧ m(a2 ) ∧ m(a3 )]∨ [t(a1 ∨ a2 ∨ a3 ∨ a4 ) ∧ m(a1 ) ∧ m(a2 ) ∧ m(a4 )] ∨ [t(a1 ∨ a2 ) ∧ m(a1 ) ∧ m(a2 )]. By an application of the lemma again we see that we can get the latter expression less than or equal to the join of terms of type A. Thus the same is also true of the original expression t(a1 ∨ a2 ∨ a3 ∨ a4 ) ∧ m(a1 ). Thus we have: Lemma 3.3. For elements a1 , a2 , . . . , an in A, we have that the expression t(a1 ∨. . .∨an )∧m(a1 ) is less than or equal to the join of terms of type A.
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DHARMANAND BABOOLAL AND PARANJOTHI PILLAY
4. The Vietoris uniformity 4.1. Construction of the uniformity. Let (A, μ) be a uniform locale. Proposition 4.1. If C ∈ μ, then = {t(c1 ∨ c2 ∨ . . . ∨ cn ) ∧ m(c1 ) ∧ . . . ∧ m(cn ) | ci ∈ C, n ∈ N} C is a cover of H(A). Proof. Since C is a cover of A, we have 1 = t(1) = t( C) = t( {∨F |F ⊆ C is finite }) = {t(∨F )|F ⊆ C is finite } the latter following because directed joins. t preserves Also 1 = m(1) = m( C) = {m(c)|c ∈ C}. Hence 1 = {t(∨F )|F ⊆ C is finite} ∧ {m(c)|c ∈ C} = {t(∨F ) ∧ m(c)|F ⊆ C is finite , c ∈ C}. Consider a typical element t(∨F ) ∧ m(c) in the above join. If the element c is not already in F , we can let F = F ∪ {c}, and then t(∨F ) ∧ m(c) ≤ t(F ) ∧ m(c). The latter term is, according to Lemma 3.3, less than or equal to the join of terms of type A. Each of these terms of type A Hence C is a cover of H(A). is in C. ∗D in H(A). Proposition 4.2. If C, D ∈ μ with C≤∗ D, then C≤ Now Cci ≤ di for some di ∈ D, and Proof. Take any t(c1 ∨ . . . ∨ cn ) ∧ m(c1 ) ∧ . . . ∧ m(cn ) ∈ C. for all i = 1, 2, . . . , n. We claim that C(t(c 1 ∨ . . . ∨ cn ) ∧ m(c1 ) . . . ∧ m(cn )) ≤ t(d1 ∨ . . . ∨ dn ) ∧ m(d1 ) ∧ . . . ∧ m(dn ) : such that Take any t(c1 ∨ . . . ∨ ck ) ∧ m(c1 ) ∧ . . . ∧ m(ck ) ∈ C t(c1 ∨ . . . ∨ ck ) ∧ m(c1 ) ∧ . . . ∧ m(ck ) ∧ t(c1 ∨ . . . ∨ cn ) ∧ m(c1 ) ∧ . . . ∧ m(cn ) = 0. For each j ∈ {1, 2, . . . , k}, cj ∧ ci = 0 for some i ∈ {1, 2, . . . , m}, otherwise there exists a j such n n that cj ∧ i=1 ci = 0. But then from the relation(iv) in Section 2 we get t( i=1 ci ) ∧ m(cj ) = 0. This is not possible. Thus for each j ∈ {1, 2, . . . , k} there exists i(j) ∈ {1, 2, . . . , n} such that cj ∧ ci(j) = 0. Thus cj ≤ di(j) . Similarly, for each i ∈ {1, 2, . . . , n} there exists j(i) ∈ {1, 2, . . . , k} such that ci ∧ cj(i) = 0. Then cj(i) ≤ di . Thus every di is above some cj(i) . Now since cj ≤ di(j) for each j = 1, 2, . . . , n, we have m(c1 ) ∧ m(c2 ) ∧ . . . ∧ m(ck ) ≤ m(di(1) ) ∧ m(di(2) ) ∧ . . . ∧ m(di(k) ). But since every di is above some cj(i) , we have m(cj(i) ) ≤ m(di ), and hence m(c1 ) ∧ m(c2 ) ∧ . . . ∧ m(ck ) ≤ m(di ) for all i. Thus m(c1 ) ∧ m(c2 ) ∧ . . . ∧ m(ck ) ≤ m(d1 ) ∧ m(d2 ) ∧ . . . ∧ m(dn ). Also each cj ≤ di(j) for each j = 1, 2, . . . , k, so c1 ∨. . .∨ck ≤ d1 ∨. . .∨dn and thus t(c1 ∨. . .∨ck ) ≤ t(d1 ∨ . . . ∨ dn ). Hence t(c1 ∨ . . . ∨ ck ) ∧ m(c1 ) ∧ m(c2 ) ∧ . . . ∧ m(ck ) ≤ t(d1 ∨ . . . ∨ dn ) ∧ m(d1 ) ∧ m(d2 ) ∧ . . . ∧ m(dn ). ∗ D. This proves the claim, and shows C≤ ≤ D. Proposition 4.3. If C, D ∈ μ and C ≤ D, then C
THE VIETORIS UNIFORMITY FOR LOCALES
7
with the ci ∈ C. Now each ci ≤ di Proof. Take any t(c1 ∨ . . . ∨ cn ) ∧ m(c1 ) ∧ . . . ∧ m(cn ) ∈ C for some di ∈ D. Thus c1 ∨ . . . ∨ cn ≤ d1 ∨ . . . ∨ dn , and hence t(c1 ∨ . . . ∨ cn ) ≤ t(d1 ∨ . . . ∨ dn ). Also m(ci ) ≤ m(di ) for each i, so m(c1 ) ∧ . . . ∧ m(cn ) ≤ m(d1 ) ∧ . . . ∧ m(dn ). Hence t(c1 ∨ . . . ∨ cn ) ∧ m(c1 ) ∧ . . . ∧ m(cn ) ≤ t(d1 ∨ . . . ∨ dn ) ∧ m(d1 ) ∧ . . . ∧ m(dn ) and the latter element belongs to D.
∧ D. Corollary 4.4. If C, D ∈ μ, then C ∧D ≤C ≤ y. Definition 4.5. We say that x y in H(A) if there exists C ∈ μ such that Cx Proposition 4.6. If a b in A, then t(a) t(b) in H(A). Proof. If a b, then there exists C ∈ μ such that Ca ≤ b. We claim that Ct(a) ≤ t(b): Suppose and t(c1 ∨ . . . ∨ cn ) ∧ m(c1 ) ∧ . . . ∧ m(cn ) ∧ t(a) = 0. that t(c1 ∨ . . . ∨ cn ) ∧ m(c1 ) ∧ . . . ∧ m(cn ) ∈ C, Now for every i we have ci ∧ a = 0, for otherwise m(ci ) ∧ t(a) = 0 for some i, and this is not possible. Thus ci ≤ b for every i, and this implies t(c1 ∨ . . . ∨ cn ) ≤ t(b). Hence t(c1 ∨ . . . ∨ cn ) ∧ m(c1 ) ∧ . . . ∧ m(cn ) ≤ t(b), thus proving the claim. Therefore t(a) t(b). Proposition 4.7. If a b in A, then m(a) m(b) in H(A). Proof. If ab, then there exists C ∈ μ such that Ca ≤ b. We claim that Cm(a) ≤ m(b): Suppose and t(c1 ∨ . . . ∨ cn ) ∧ m(c1 ) ∧ . . . ∧ m(cn ) ∧ m(a) = 0. that t(c1 ∨ . . . ∨ cn ) ∧ m(c1 ) ∧ . . . ∧ m(cn ) ∈ C, n Now n there must exist i such that ci ∧ a = 0. If not, then i=1 ci ∧ a = 0. But this implies t( i=1 ci ) ∧ m(a) = 0, which is not possible. Thus ci ≤ b. Hence m(ci ) ≤ m(b), and so t(c1 ∨ . . . ∨ cn ) ∧ m(c1 ) ∧ . . . ∧ m(cn ) ≤ m(ci ) ≤ m(b). This proves the claim. Therefore m(a) m(b). Proposition 4.8. For x, y, z ∈ H(A), the relation satisfies: (i) x, y z ⇒ x ∨ y z. (ii) z x, y ⇒ z x ∧ y. ≤ z and Dy ≤ z. Put E = C ∧D ∈ μ. Proof. (i) If x, yz, then we can find C, D ∈ μ such that Cx ≤ z and Ey ≤ z. Then E ≤ C and E ≤ D, so by Proposition 4.3, E ≤ C and E ≤ D. Thus Ex This implies E(x ∨ y) ≤ z, so x ∨ y z.
≤ x and Dz ≤ y. Put E = C ∧D ∈ μ. (ii) If z x, y, then then we can find C, D ∈ μ such that Cz ≤ x ∧ y, so z x ∧ y. Then as in (i) above we can get Ez ≤ x and Ez ≤ y. This implies Ez Proposition 4.9. If a a , and bi bi for i = 1, 2, . . . , n in A, then t(a) ∧ m(b1 ) ∧ . . . ∧ m(bn ) t(a ) ∧ m(b1 ) ∧ . . . ∧ m(bn ) in H(A).
Proof. This follows from the last three propositions.
Proposition 4.10. The relation is an admissible relation on H(A), i.e. for each x ∈ H(A), x = y(y x). Proof. Take any basicgenerator t(a) ∧ m(b1 ) ∧ . . . ∧ m(bn ) of H(A). Now a = a (a a), and foreach i, bi = bi (bi bi ). Since the collection{a ∈ A|a a} is updirected, we have t(a) = t(a )(a a). Also for each i we have m(bi ) = m(bi )(bi bi ). Thus t(a) ∧ m(b1 ) ∧ . . . ∧ m(bn ) = t(a )(a a) ∧ m(b1 )(b1 b1 ) ∧ . . . ∧ m(bn )(bn bn ) = {t(a ) ∧ m(b1 ) ∧ . . . ∧ m(bn )|a a, b1 b1 , . . . , bn bn }. From the previous proposition t(a ) ∧ m(b1 ) ∧ . . . ∧ m(bn ) t(a) ∧ m(b1 ) ∧ . . . ∧ m(bn ). From this we conclude that is an admissible relation.
8
DHARMANAND BABOOLAL AND PARANJOTHI PILLAY
From the above sequence of propositions we obtain: Theorem 4.11. Let (A, μ) be a uniform locale, and let H(A) be the Vietoris locale of A. The ∈ μ} forms a basis for a uniformity μ collection {C|C on H(A). We will refer to (H(A), μ ) as the Vietoris uniform locale associated with the uniform locale (A, μ), and to the uniformity on H(A) as the Vietoris uniformity. It is clear that if the uniform frame (A, μ) has a countable uniformity basis, then so does the Vietoris uniform frame (H(A), μ ). Thus we have Proposition 4.12. If a uniform frame is metrizable, then so is its Vietoris uniform frame. 4.2. The Vietoris uniformity functor. In the next proposition the map g refers to the map introduced in diagram (3.1.1), and in the second proposition this map is referred to as gA and gB since we are dealing there with the two uniform frames A and B. Proposition 4.13. The map g : H(A) → A is uniform and a surjection. = {t(c1 ∨ c2 ∨ ... ∨ cn ) ∧ m(c1 ) ∧ m(c2 ) ∧ ... ∧ m(cn )|ci ∈ C, n ∈ N} any basic Proof. Take C uniform cover of H(A). Then = {g(t(c1 ∨ c2 ∨ ... ∨ cn ) ∧ m(c1 ) ∧ m(c2 ) ∧ ... ∧ m(cn ))|ci ∈ C, n ∈ N} g(C) = {c1 ∧ c2 ∧ ... ∧ cn |ci ∈ C, n ∈ N}. so g(C) ∈ μ and hence g is uniform. The map g is onto as we saw earlier, and Now C ≤ g(C), ≤ C, so g is also a surjection. for any C ∈ μ we have g(C) Proposition 4.14. For any frame map f : A → B between frames, there exists a frame homomorphism f˜ from H(A) to H(B) such that f˜(m(a)) = m(f (a)) and f˜(t(a)) = t(f (a)) for each a ∈ A, and making the following diagram commutative: H(A)
f˜
gA
H(B) gB
f
A
B
Proof. Let XA be the set consisting of all symbols t(a) and m(a) as a varies over A, and likewise for the set XB . Let F XA and F XB be the free frames respectively, and let νA and νB be the corresponding quotient maps defined earlier. Consider the diagram XA
iA
F XA
νA
F XA /ΦA = H(A)
p
XB
f˜
ψ iB
F XB
νB
H(B)
where p is the map m(a) → m(f (a)), t(a) → t(f (a)), and ψ is the unique frame homomorphism such that ψ ◦ iA = iB ◦ p. Thus ψ(m(a)) = m(f (a) and ψ(t(a)) = t(f (a) for all a ∈ A. It can be verified that the relation RA on F XA that determines H(A) is such that RA ⊆ ker(νB ◦ ψ). Thus there exists a unique frame homomorphism f˜: H(A) → H(B) such that f˜ ◦ νA = νB ◦ ψ. If we suppress the quotient maps and think of the m(a) and the t(a) as elements of the Vietoris frame, then we get f˜(m(a)) = m(f (a)) and f˜(t(a)) = t(f (a)) for each a ∈ A. Finally, for any a ∈ A we have (gB ◦ f˜)(m(a)) = gB (f˜(m(a))) = gB (m(f (a)) = f (a), and (f ◦ gA )(m(a)) = f (gA (m(a))) = f (a). A similar calculation works for the t(a). Thus gB ◦ f˜ agrees with f ◦ gA on the generators of H(A) and hence they are equal.
THE VIETORIS UNIFORMITY FOR LOCALES
9
Proposition 4.15. If A and B are uniform frames and f : A → B is a uniform frame map, then so is f˜: H(A) → H(B). Proof. Take any basic uniform cover C˜ of H(A), where C is a uniform cover of A. Now f (C) is ˜ which then implies f˜(C) ˜ is a uniform a uniform cover of B. We show that in fact f (C) = f˜(C), cover. For this take any t(f (c1 ) ∨ ... ∨ f (cn )) ∧ m(f (c1 )) ∧ ... ∧ m(f (cn )) ∈ f (C, and note that t(f (c1 ) ∨ ... ∨ f (cn )) ∧ m(f (c1 )) ∧ ... ∧ m(f (cn )) = t(f (c1 ∨ ... ∨ cn )) ∧ m(f (c1 )) ∧ ... ∧ m(f (cn )) = f˜(t(c1 ∨ ... ∨ cn )) ∧ f˜(m(c1 )) ∧ ... ∧ f˜(m(cn )) = f˜(t(c1 ∨ ... ∨ cn ) ∧ m(c1 ) ∧ ... ∧ m(cn )). Thus f˜ is uniform.
Observe that if we have a diagram f
h
→B− →C A− ˜ ◦ f˜: For any a ∈ A we have in UniFrm then h ◦f =h ˜ ◦ f˜)(m(a)) = h(m(f ˜ (h (a)) = m(h(f (a)) = (h ◦ f )(m(a)), ˜ ◦ f˜. and the same for the t(a). Since they agree on the generators of H(A) we have h ◦f = h is the identity on H(A). Thus Similarly one can see that if id : A → A is the identity, then id we have: Theorem 4.16. The assignment of a uniform frame A to its Vietoris uniform frame H(A), and of a uniform frame map f : A → B to the uniform map f˜: H(A) → H(B) makes H : UniFrm → UniFrm a functor. 4.3. Completion. Theorem 4.17. If A is a uniform frame such that H(A) is complete, then A must be complete. Proof. As we saw earlier, A is embedded as a closed uniform sublocale of H(A). Since closed uniform sublocales of complete uniform locales are complete in the induced uniformity ([1]), the result follows. In the classical theory of hyperspaces, Morita ([8]) showed that if a uniform space is complete then its hyperspace of non-empty compact subsets with its hyperspace uniformity must be complete. We think that the result is also true in the pointfree setting, but we have not been able to prove this. We conclude with this open question. Question. If a uniform frame A is complete, is it true that H(A) is complete? Acknowledgements. We are grateful to Aleˇs Pultr for the many valuable discussions relating to this work. We also thank the referee for helpful comments and suggestions which allowed us to improve the presentation of our paper. References [1] B. Banaschewski, Uniform completion in pointfree topology, in: Topological and Algebraic Structures in Fuzzy Sets, pp. 19-56. Trends Log. Stud. Log. Libr., vol. 20, Kluwer Academic Publishers, Boston, Dordrecht, London, 2003. [2] B. Banaschewski and A. Pultr, Samuel compactification and completion of uniform frames, Math. Proc. Camb. Phil. Soc. 108 (1990), 63-78. [3] A. Illanes and S. B. Nadler, Hyperspaces, Marcel Dekker (1999). [4] J. R. Isbell, Uniform Spaces, Amer. Math. Soc., Providence (1964). [5] P. T. Johnstone, Stone Spaces, Cambridge University Press (1982).
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DHARMANAND BABOOLAL AND PARANJOTHI PILLAY
[6] P. T. Johnstone, Vietoris Locales and Localic Semilattices, Continuous lattices and their applications (Bremen, 1982), 155-180, Lecture Notes in Pure and Appl. Math., 101, Dekker, New York (1985). [7] E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182. [8] K. Morita, Completion of hyperspaces of compact subsets and topological completion of open-closed maps, General Topology and Appl. 4 (1974), 217-233. [9] J. Picado and A. Pultr, Frames and Locales, Springer, Basel (2012). School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa Email address: [email protected], [email protected]
Dharmanand Baboolal, Paranjothi Pillay
PII:
S0166-8641(19)30384-0
DOI:
https://doi.org/10.1016/j.topol.2019.106979
Reference:
TOPOL 106979
To appear in:
Topology and its Applications
Received date:
9 December 2018
Revised date:
29 March 2019
Accepted date:
1 April 2019
Please cite this article as: D. Baboolal, P. Pillay, The Vietoris uniformity for locales, Topol. Appl. (2019), 106979, doi: https://doi.org/10.1016/j.topol.2019.106979.
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THE VIETORIS UNIFORMITY FOR LOCALES DHARMANAND BABOOLAL AND PARANJOTHI PILLAY Dedicated to Aleˇs Pultr on his 80th birthday
Abstract. The purpose of this paper is to define the Vietoris uniformity on the Vietoris locale of a uniform locale A. This is the pointfree counterpart to the classical construction of the hyperspace uniformity on the hyperspace of non-empty compact subsets of a uniform space.
1. Introduction The purpose of this paper is to define the Vietoris uniformity on the Vietoris locale of a uniform locale A. The Vietoris locale V (A) of an arbitrary locale A was defined by Johnstone [6], and it is the analogue of the hyperspace of all compact subsets, including the empty set, of an arbitrary topological space (see e.g. [3], [7]). However, we will focus specifically on the pointfree analogue of the hyperspace of all non-empty compact subsets of a space. This amounts to working with a specific closed sublocale of V (A), what Johnstone refers to as V + (A) in his paper. We do this because in the classical theory of hyperspaces one works only with the non-empty subsets of a space, and we wanted to capture this theory in the pointfree context. To simplify the notation we will rename V + (A) as H(A) throughout the paper. One of the things considered by Johnstone in his paper was to see which properties possessed by A would be inherited by V (A). He showed this is true for the properties of regularity, complete regularity and 0-dimensionality, amongst others. Our paper can be seen as an extension of this idea in the sense that if we start with a uniformity on A we show that this uniformity can be extended to a uniformity on H(A), i.e. A can be uniformly embedded in H(A), and in fact as a closed sublocale of H(A). We also show that this construction makes H functorial. We also show that if H(A) is complete then so is A, but we leave as an open question whether the converse is also true. 2. Preliminaries A frame A is a complete lattice which satisfies the infinite distributive law: x ∧ S = {x ∧ s|s ∈ S} for all x ∈ A, S ⊆ A. The top element of A is denoted by 1 and the bottom by 0. A frame homomorphism is a map f : A → B between frames that preserves finitary meets (including the top 1) and arbitrary joins (including the bottom 0). We thus have the category of frames and frame homomorphisms, which we denote by Frm. A frame map f : A → B is said to be dense if x = 0 whenever f (x) = 0. Recall that every frame homomorphism f : A → B has a right adjoint f∗ : B → A characterized by the condition f (x) ≤ y ⇔ x ≤ f∗ (y). Date: 30 March 2019. 1991 Mathematics Subject Classification. 06D22, 54D35, 54E15. Key words and phrases. frame, locale, Vietoris locale, uniformity, completion. A grant from the National Research Foundation (S.A), Grant No. 113840, is gratefully acknowledged. 1
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DHARMANAND BABOOLAL AND PARANJOTHI PILLAY
If f is onto then f∗ is given by the formula f∗ (y) = {x ∈ A|f (x) = y}. It follows that for such f the composite map f f∗ is the identity map. For every pair of elements a, b in a frame we have the element a → b given by the Heyting operation → characterised by the condition x ≤ a → b iff x ∧ a ≤ b. The category Loc is the category with objects frames (or locales) and with morphisms f : A → B where f is the right adjoint of some frame map g : B → A. Sublocales of a locale A are the regular subobjects of A in Loc, or in Frm the quotients of A. Sublocales are those subsets S of A having the characteristic properties: (1) If M ⊆ S then M ∈ S. (2) If a ∈ A, s ∈ S then a → s ∈ S. The collection of sublocales S(A) of a locale A ordered by inclusion form a co-frame, that is, Ti = (S ∨ Ti ) S∨ i∈I
i∈I
for sublocales S and Ti . Here, the infimum of sublocales is just theset-theoretic intersection, and the supremum of a collection of sublocales Ti is the sublocale { M |M ⊆ Ti }. An open sublocale is one of the form o(a) = {a → x|x ∈ A} = {x ∈ A|x = a → x}, and a closed sublocale is one of the form c(a) =↑ a. The sublocales o(a) and c(a) are complements of each other in the sublocale lattice. We also have o(
o(0) =O i ai ) = i o(ai )
↑(
A o(1) = i ai ) = i ↑ ai
o(a ∧ b) = o(a) ∩ o(b) ↑ (a ∧ b) =↑ a ∨ ↑ b
where O = {1} is the trivial sublocale. A frame homomorphism f : A → B is said to be closed if f∗ (b ∨ f (a)) = f∗ (b) ∨ a for all a ∈ A and all b ∈ B. This is equivalent to saying that the localic map f∗ maps closed sublocales of B to closed sublocales of A. For elements a, b ∈ A, we say that a is rather below b, written a ≺ b, if there exists an element c ∈ A such that a ∧ c = 0 and c ∨ b = 1. This is equivalent to the condition that a∗ ∨ b = 1, where a∗ is the pseudocomplement a → 0 of a, i.e. the largest element in A whose meet with a is 0. A frame A is called regular if for each a ∈ A, a = {x ∈ A|x ≺ a}. A frame A is called completely regular if for each a ∈ A, a = {x ∈ A|x ≺≺ a} where x ≺≺ a means that there exists a doubly indexed sequence of elements in A, (xnk )n=0,1,...;k=0,1,...,2n such that x = xn0 , xnk ≺ xnk+1 , xn2n = a, xnk = xn+1 2k for all n = 0, 1, ... and k = 0, 1, ..., 2n . A frame A is said to be 0-dimensional if it has a basis consisting of complemented elements, i.e. elements a such that a ∨ a∗ = 1. A cover of a frame A is a subset C of A such that C = 1. A cover C is said to refine a cover D (written as C ≤ D) if for each c ∈ C there exists d ∈ D such that c ≤ d. A cover ∗ C is said to star-refine a cover D (written as C ≤ D) if the cover {Cx|x ∈ C} ≤ D, where Cx = {c ∈ C|c ∧ x = 0}. A uniform frame (A, μ) is a frame A together with a specified collection μ of covers, called uniform covers, satisfying the following properties: (i) If C ∈ μ and C ≤ D, then D ∈ μ. (ii) If C, D ∈ μ, then C ∧ D ∈ μ, where C ∧ D = {c ∧ d|c ∈ C, d ∈ D}. (iii) If C ∈ μ then there exists D ∈μ such that D ≤∗ C. (iv) For each a ∈ A we have a = x(x a), where x a means that there exists some C ∈ μ such that Cx ≤ a. A uniformity basis μ0 for a uniformity μ on A is a subcollection of μ such that for each C ∈ μ there exists a D ∈ μ0 such that D ≤ C. A uniform frame is said to be of countable type if the uniform covers have a countable basis.
THE VIETORIS UNIFORMITY FOR LOCALES
3
A frame map f : (A, μ) → (B, ν) between uniform frames is called uniform if f (C) ∈ ν whenever C ∈ μ. If, furthermore, f is onto and for each D ∈ ν there exists C ∈ μ such that f (C) ≤ D, then we say that f is a surjection. We also say in this situation that B is uniformly embedded in A. The uniform frame (B, ν) is said to be complete if whenever (A, μ) is a uniform frame such that f : (A, μ) → (B, ν) is a dense surjection, then f is an isomorphism. The category of uniform frames and uniform frame maps will be denoted by UniFrm. For general background on frames the reader is referred to the books [5] and [9]. 3. The Vietoris Locale To make this paper self-contained we recall Johnstone’s construction of the Vietoris locale and some of its basic facts (see [6]). Let A be a locale. For each a ∈ A, let t(a) and m(a) be abstract symbols. Then V (A) is the frame freely generated by these symbols subject to the following relations: t(a ∧ b) = t(a) ∧ t(b) for all a, b ∈ A, and t(1) = 1. t( S) = t(s)(s ∈ S) for all updirected S ⊆ A. m( S) = m(s)(s ∈ S) for all S ⊆ A, and m(0) = 0.
(iii)
m(a ∧ b) ≥ t(a) ∧ m(b) for all a, b ∈ A.
(iv)
t(a ∨ b) ≤ t(a) ∨ m(b) for all a, b ∈ A.
(v)
(i) (ii)
As Johnstone remarks in his paper one can think informally of V (A) as the space of all compact subspaces of A, of t(a) as the set of those compact subspaces contained in a, and of m(a) as the set of those compact subspaces that meet a. Let X = {m(a), t(a)|a ∈ A}. The above relations give the construction of V (A) diagrammatically as shown below: X
FX f
f¯
ν
F X/Θ = V (A) ϕ
A Here f is the map t(a) −→ a, and m(a) −→ a, F X is the free frame on X, X → F X is the insertion of generators, and f¯ is the unique frame homomorphism making the left triangle in the diagram commute. It is easily verified that the relation R on F X determined by the relations (i)-(v) is such that R ⊆ ker f¯. Thus if Θ is the congruence on F X generated by R, and ν the corresponding quotient map, we obtain a unique frame homomorphism ϕ : F X/Θ → A making the second triangle in the diagram commute. The Vietoris locale V (A) is defined to be the frame F X/Θ. (See [9, IV.2] for a discussion on free frame constructions). 3.1. Some properties of V (A). If w ∈ F X, then ν(w) ∈ V (A). We will write ν(w) as w, i.e. we will suppress the quotient map ν. Bearing this in mind, we then see that ϕ(t(a)) = a and ϕ(m(a)) = a. Hence: (a) ϕ : V (A) → A is an onto frame homomorphism. Thus, A is a sublocale of V (A). If A is regular, then V (A) is regular and in this case ϕ : V (A) → A is also closed. Thus A is embedded as a closed sublocale of V (A) ([6]). (b) t(0) and m(1) are complements of each other in V (A). To see this, note from the relation (iv) above that t(0) ∧ m(1) ≤ m(0 ∧ 1) = m(0) = 0. Thus t(0) ∧ m(1) = 0. Also 1 = t(1) = t(0 ∨ 1) ≤ t(0) ∨ m(1), hence t(0) ∨ m(1) = 1. Of course m(0) and t(1) are also complements of each other in V (A) ([6]).
4
DHARMANAND BABOOLAL AND PARANJOTHI PILLAY
(c) Every element x of V (A) is a join of elements of the type t(a1 ) ∧ t(a2 ) ∧ . . . ∧ t(am ) ∧ m(b1 ) ∧ m(b2 ) ∧ . . . ∧ m(bn ). Since t preserves finite meet, a basic generator has the form t(a) ∧ m(b1 ) ∧ m(b2 ) ∧ . . . ∧ m(bn ). But from relation (iv) of the definition of V (A) we have t(a) ∧ m(b1 ) ∧ m(b2 ) ∧ . . . ∧ m(bn ) = t(a) ∧ m(a ∧ b1 ) ∧ m(a ∧ b2 ) ∧ . . . ∧ m(a ∧ bn ). Thus the basic generators have the form t(a) ∧ m(b1 ) ∧ m(b2 ) ∧ . . . ∧ m(bn ) where bi ≤ a for each i ([6]). (d) From (b) above we see that V (A) can be written as the disjoint join of the two closed sublocales ↑ m(1) and ↑ t(0), what is referred to by Johnstone as V0 (A) and V + (A) respectively. Of course these sublocales are also open, hence they are clopen ([6]). (e) If one adjoins the relation t(0) = 1 to those relations (i)-(v) that define V (A), then one gets the sublocale V0 (A). This is because t(0) = 1 ⇔ m(1) = 0, since t(0) and m(1) are complementary. It is shown in [6] that V0 (A) ∼ = 2, where 2 is the terminal object in Loc. (f) If one adjoins the relation t(0) = 0 to those relations (i)-(v) that define V (A), then one gets the sublocale V + (A) of V (A). The sublocale V + (A) corresponds to the hyperspace of all non-empty compact subsets in the setting of spaces. Since V + (A) would be our primary interest of study from now on, it may be better to change notation and refer to V + (A) as H(A). Thus H(A) is the Vietoris locale of all “non-empty compact subspaces” of A. The relations (i)-(v) as well as (vi) t(0) = 0, then determine H(A), and we can represent this diagrammatically as X
FX f
f¯
ν
F X/Φ = H(A) g
(3.1.1)
A Just as before, g is a frame homomorphism which is onto since g(m(a)) = a and g(t(a)) = a for all a ∈ A. The map g is also closed if A is regular. This is because the map ϕ : V (A) → A was shown to be closed for such A ([6]), and H(A) has one more defining relation than V (A). 3.2. Some properties of H(A). The extra relation t(0) = 0 means that H(A) satisfies some properties not enjoyed by V (A). We list these useful properties which are of crucial importance in the sequel. (a) t(0) = 0 ⇔ t(a) ≤ m(a) for all a ∈ A. To see this note that t(a) = t(0 ∨ a) ≤ t(0) ∨ m(a) = 0 ∨ m(a) = m(a). For the other direction, if t(a) ≤ m(a) for all a, then t(0) ≤ m(0) = 0. Hence t(0) = 0. (b) m(1) = 1. This follows since, as we saw before, t(0) and m(1) are complementary. Proposition 3.1. In H(A), the collection T = {t(a) ∧ m(b1 ) ∧ m(b2 ) ∧ . . . ∧ m(bn )| a = b1 ∨ b2 ∨ . . . ∨ bn , a ∈ A, bi ∈ A, n ∈ N} is a basis for H(A).
THE VIETORIS UNIFORMITY FOR LOCALES
5
Proof. Take a basic generator of H(A), say, t(a) ∧ m(b1 ) ∧ m(b2 ) ∧ . . . ∧ m(bn ) with bi ≤ a for all i. Since t(a) ≤ m(a) in H(A), we have t(a) ∧ m(b1 ) ∧ m(b2 ) ∧ . . . ∧ m(bn ) = t(a) ∧ m(a) ∧ m(b1 ) ∧ m(b2 ) ∧ . . . ∧ m(bn ) with a ∨ b1 ∨ b2 ∨ . . . ∨ bn = a, and the latter is in T .
The following lemma will be useful in the section on the Vietoris uniformity. Lemma 3.2. For elements a1 , a2 , . . . , an , an+1 in A, we have: t(a1 ∨ . . . ∨ an+1 ) ∧ m(a1 ) ∧ . . . ∧ m(an ) ≤ ≤[t(a1 ∨ . . . ∨ an+1 ) ∧ (m(a1 ) ∧ m(a2 ) ∧ . . . m(an+1 ))] ∨ [t(a1 ∨ . . . ∨ an ) ∧ (m(a1 ) ∧ . . . ∧ m(an ))]. Proof. The left hand side is less than or equal to [t(a1 ∨ . . . ∨ an ) ∨ m(an+1 )] ∧ [m(a1 ) ∧ . . . ∧ m(an )] = [t(a1 ∨ . . . ∨ an ) ∧ m(a1 ) ∧ . . . ∧ m(an )] ∨ [m(a1 ) ∧ . . . ∧ m(an+1 )] ≤ t(a1 ∨ . . . ∨ an+1 ) ∧ {[t(a1 ∨ . . . ∨ an ) ∧ m(a1 ) ∧ . . . ∧ m(an )] ∨ [m(a1 ) ∧ . . . ∧ m(an+1 )]} = [t(a1 ∨ . . . ∨ an ) ∧ m(a1 ) ∧ . . . ∧ m(an )] ∨ [t(a1 ∨ . . . ∨ an+1 ) ∧ m(a1 ) ∧ . . . ∧ m(an+1 )].
We note that the two terms in square brackets that appear in the last line in the above proof each have the following property: The join of the arguments of m is the argument of t. We shall refer to these terms as terms of type A. The above lemma allows us to make some computations: t(a1 ∨ a2 ) ∧ m(a1 ) ≤ [t(a1 ∨ a2 ) ∧ m(a1 ) ∧ m(a2 )] ∨ [t(a1 ) ∧ m(a1 )]. Thus the left hand side of the above expression is less than or equal to the join of terms of type A. For a little more involved expression: t(a1 ∨ a2 ∨ a3 ∨ a4 ) ∧ m(a1 ) ≤ [t(a1 ∨ a2 ∨ a3 ∨ a4 ) ∧ m(a1 ) ∧ m(a2 ∨ a3 ∨ a4 )] ∨ [t(a1 ) ∧ m(a1 )] = [t(a1 ∨ a2 ∨ a3 ∨ a4 ) ∧ m(a1 ) ∧ (m(a2 ) ∨ m(a3 ) ∨ m(a4 ))] ∨ [t(a1 ) ∧ m(a1 )] = [t(a1 ∨ a2 ∨ a3 ∨ a4 ) ∧ m(a1 ) ∧ m(a2 )] ∨ . . . ∨ . . . ∨ [t(a1 ) ∧ m(a1 )]. If we consider the expression t(a1 ∨ a2 ∨ a3 ∨ a4 ) ∧ m(a1 ) ∧ m(a2 ), we see that we can use the above lemma to get this expression to be less than or equal to [t(a1 ∨ a2 ∨ a3 ∨ a4 ) ∧ m(a1 ) ∧ m(a2 ) ∧ m(a3 ∨ a4 )] ∨ [t(a1 ∨ a2 ) ∧ m(a1 ) ∧ m(a2 )] and this is equal to [t(a1 ∨ a2 ∨ a3 ∨ a4 ) ∧ m(a1 ) ∧ m(a2 ) ∧ m(a3 )]∨ [t(a1 ∨ a2 ∨ a3 ∨ a4 ) ∧ m(a1 ) ∧ m(a2 ) ∧ m(a4 )] ∨ [t(a1 ∨ a2 ) ∧ m(a1 ) ∧ m(a2 )]. By an application of the lemma again we see that we can get the latter expression less than or equal to the join of terms of type A. Thus the same is also true of the original expression t(a1 ∨ a2 ∨ a3 ∨ a4 ) ∧ m(a1 ). Thus we have: Lemma 3.3. For elements a1 , a2 , . . . , an in A, we have that the expression t(a1 ∨. . .∨an )∧m(a1 ) is less than or equal to the join of terms of type A.
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DHARMANAND BABOOLAL AND PARANJOTHI PILLAY
4. The Vietoris uniformity 4.1. Construction of the uniformity. Let (A, μ) be a uniform locale. Proposition 4.1. If C ∈ μ, then = {t(c1 ∨ c2 ∨ . . . ∨ cn ) ∧ m(c1 ) ∧ . . . ∧ m(cn ) | ci ∈ C, n ∈ N} C is a cover of H(A). Proof. Since C is a cover of A, we have 1 = t(1) = t( C) = t( {∨F |F ⊆ C is finite }) = {t(∨F )|F ⊆ C is finite } the latter following because directed joins. t preserves Also 1 = m(1) = m( C) = {m(c)|c ∈ C}. Hence 1 = {t(∨F )|F ⊆ C is finite} ∧ {m(c)|c ∈ C} = {t(∨F ) ∧ m(c)|F ⊆ C is finite , c ∈ C}. Consider a typical element t(∨F ) ∧ m(c) in the above join. If the element c is not already in F , we can let F = F ∪ {c}, and then t(∨F ) ∧ m(c) ≤ t(F ) ∧ m(c). The latter term is, according to Lemma 3.3, less than or equal to the join of terms of type A. Each of these terms of type A Hence C is a cover of H(A). is in C. ∗D in H(A). Proposition 4.2. If C, D ∈ μ with C≤∗ D, then C≤ Now Cci ≤ di for some di ∈ D, and Proof. Take any t(c1 ∨ . . . ∨ cn ) ∧ m(c1 ) ∧ . . . ∧ m(cn ) ∈ C. for all i = 1, 2, . . . , n. We claim that C(t(c 1 ∨ . . . ∨ cn ) ∧ m(c1 ) . . . ∧ m(cn )) ≤ t(d1 ∨ . . . ∨ dn ) ∧ m(d1 ) ∧ . . . ∧ m(dn ) : such that Take any t(c1 ∨ . . . ∨ ck ) ∧ m(c1 ) ∧ . . . ∧ m(ck ) ∈ C t(c1 ∨ . . . ∨ ck ) ∧ m(c1 ) ∧ . . . ∧ m(ck ) ∧ t(c1 ∨ . . . ∨ cn ) ∧ m(c1 ) ∧ . . . ∧ m(cn ) = 0. For each j ∈ {1, 2, . . . , k}, cj ∧ ci = 0 for some i ∈ {1, 2, . . . , m}, otherwise there exists a j such n n that cj ∧ i=1 ci = 0. But then from the relation(iv) in Section 2 we get t( i=1 ci ) ∧ m(cj ) = 0. This is not possible. Thus for each j ∈ {1, 2, . . . , k} there exists i(j) ∈ {1, 2, . . . , n} such that cj ∧ ci(j) = 0. Thus cj ≤ di(j) . Similarly, for each i ∈ {1, 2, . . . , n} there exists j(i) ∈ {1, 2, . . . , k} such that ci ∧ cj(i) = 0. Then cj(i) ≤ di . Thus every di is above some cj(i) . Now since cj ≤ di(j) for each j = 1, 2, . . . , n, we have m(c1 ) ∧ m(c2 ) ∧ . . . ∧ m(ck ) ≤ m(di(1) ) ∧ m(di(2) ) ∧ . . . ∧ m(di(k) ). But since every di is above some cj(i) , we have m(cj(i) ) ≤ m(di ), and hence m(c1 ) ∧ m(c2 ) ∧ . . . ∧ m(ck ) ≤ m(di ) for all i. Thus m(c1 ) ∧ m(c2 ) ∧ . . . ∧ m(ck ) ≤ m(d1 ) ∧ m(d2 ) ∧ . . . ∧ m(dn ). Also each cj ≤ di(j) for each j = 1, 2, . . . , k, so c1 ∨. . .∨ck ≤ d1 ∨. . .∨dn and thus t(c1 ∨. . .∨ck ) ≤ t(d1 ∨ . . . ∨ dn ). Hence t(c1 ∨ . . . ∨ ck ) ∧ m(c1 ) ∧ m(c2 ) ∧ . . . ∧ m(ck ) ≤ t(d1 ∨ . . . ∨ dn ) ∧ m(d1 ) ∧ m(d2 ) ∧ . . . ∧ m(dn ). ∗ D. This proves the claim, and shows C≤ ≤ D. Proposition 4.3. If C, D ∈ μ and C ≤ D, then C
THE VIETORIS UNIFORMITY FOR LOCALES
7
with the ci ∈ C. Now each ci ≤ di Proof. Take any t(c1 ∨ . . . ∨ cn ) ∧ m(c1 ) ∧ . . . ∧ m(cn ) ∈ C for some di ∈ D. Thus c1 ∨ . . . ∨ cn ≤ d1 ∨ . . . ∨ dn , and hence t(c1 ∨ . . . ∨ cn ) ≤ t(d1 ∨ . . . ∨ dn ). Also m(ci ) ≤ m(di ) for each i, so m(c1 ) ∧ . . . ∧ m(cn ) ≤ m(d1 ) ∧ . . . ∧ m(dn ). Hence t(c1 ∨ . . . ∨ cn ) ∧ m(c1 ) ∧ . . . ∧ m(cn ) ≤ t(d1 ∨ . . . ∨ dn ) ∧ m(d1 ) ∧ . . . ∧ m(dn ) and the latter element belongs to D.
∧ D. Corollary 4.4. If C, D ∈ μ, then C ∧D ≤C ≤ y. Definition 4.5. We say that x y in H(A) if there exists C ∈ μ such that Cx Proposition 4.6. If a b in A, then t(a) t(b) in H(A). Proof. If a b, then there exists C ∈ μ such that Ca ≤ b. We claim that Ct(a) ≤ t(b): Suppose and t(c1 ∨ . . . ∨ cn ) ∧ m(c1 ) ∧ . . . ∧ m(cn ) ∧ t(a) = 0. that t(c1 ∨ . . . ∨ cn ) ∧ m(c1 ) ∧ . . . ∧ m(cn ) ∈ C, Now for every i we have ci ∧ a = 0, for otherwise m(ci ) ∧ t(a) = 0 for some i, and this is not possible. Thus ci ≤ b for every i, and this implies t(c1 ∨ . . . ∨ cn ) ≤ t(b). Hence t(c1 ∨ . . . ∨ cn ) ∧ m(c1 ) ∧ . . . ∧ m(cn ) ≤ t(b), thus proving the claim. Therefore t(a) t(b). Proposition 4.7. If a b in A, then m(a) m(b) in H(A). Proof. If ab, then there exists C ∈ μ such that Ca ≤ b. We claim that Cm(a) ≤ m(b): Suppose and t(c1 ∨ . . . ∨ cn ) ∧ m(c1 ) ∧ . . . ∧ m(cn ) ∧ m(a) = 0. that t(c1 ∨ . . . ∨ cn ) ∧ m(c1 ) ∧ . . . ∧ m(cn ) ∈ C, n Now n there must exist i such that ci ∧ a = 0. If not, then i=1 ci ∧ a = 0. But this implies t( i=1 ci ) ∧ m(a) = 0, which is not possible. Thus ci ≤ b. Hence m(ci ) ≤ m(b), and so t(c1 ∨ . . . ∨ cn ) ∧ m(c1 ) ∧ . . . ∧ m(cn ) ≤ m(ci ) ≤ m(b). This proves the claim. Therefore m(a) m(b). Proposition 4.8. For x, y, z ∈ H(A), the relation satisfies: (i) x, y z ⇒ x ∨ y z. (ii) z x, y ⇒ z x ∧ y. ≤ z and Dy ≤ z. Put E = C ∧D ∈ μ. Proof. (i) If x, yz, then we can find C, D ∈ μ such that Cx ≤ z and Ey ≤ z. Then E ≤ C and E ≤ D, so by Proposition 4.3, E ≤ C and E ≤ D. Thus Ex This implies E(x ∨ y) ≤ z, so x ∨ y z.
≤ x and Dz ≤ y. Put E = C ∧D ∈ μ. (ii) If z x, y, then then we can find C, D ∈ μ such that Cz ≤ x ∧ y, so z x ∧ y. Then as in (i) above we can get Ez ≤ x and Ez ≤ y. This implies Ez Proposition 4.9. If a a , and bi bi for i = 1, 2, . . . , n in A, then t(a) ∧ m(b1 ) ∧ . . . ∧ m(bn ) t(a ) ∧ m(b1 ) ∧ . . . ∧ m(bn ) in H(A).
Proof. This follows from the last three propositions.
Proposition 4.10. The relation is an admissible relation on H(A), i.e. for each x ∈ H(A), x = y(y x). Proof. Take any basicgenerator t(a) ∧ m(b1 ) ∧ . . . ∧ m(bn ) of H(A). Now a = a (a a), and foreach i, bi = bi (bi bi ). Since the collection{a ∈ A|a a} is updirected, we have t(a) = t(a )(a a). Also for each i we have m(bi ) = m(bi )(bi bi ). Thus t(a) ∧ m(b1 ) ∧ . . . ∧ m(bn ) = t(a )(a a) ∧ m(b1 )(b1 b1 ) ∧ . . . ∧ m(bn )(bn bn ) = {t(a ) ∧ m(b1 ) ∧ . . . ∧ m(bn )|a a, b1 b1 , . . . , bn bn }. From the previous proposition t(a ) ∧ m(b1 ) ∧ . . . ∧ m(bn ) t(a) ∧ m(b1 ) ∧ . . . ∧ m(bn ). From this we conclude that is an admissible relation.
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DHARMANAND BABOOLAL AND PARANJOTHI PILLAY
From the above sequence of propositions we obtain: Theorem 4.11. Let (A, μ) be a uniform locale, and let H(A) be the Vietoris locale of A. The ∈ μ} forms a basis for a uniformity μ collection {C|C on H(A). We will refer to (H(A), μ ) as the Vietoris uniform locale associated with the uniform locale (A, μ), and to the uniformity on H(A) as the Vietoris uniformity. It is clear that if the uniform frame (A, μ) has a countable uniformity basis, then so does the Vietoris uniform frame (H(A), μ ). Thus we have Proposition 4.12. If a uniform frame is metrizable, then so is its Vietoris uniform frame. 4.2. The Vietoris uniformity functor. In the next proposition the map g refers to the map introduced in diagram (3.1.1), and in the second proposition this map is referred to as gA and gB since we are dealing there with the two uniform frames A and B. Proposition 4.13. The map g : H(A) → A is uniform and a surjection. = {t(c1 ∨ c2 ∨ ... ∨ cn ) ∧ m(c1 ) ∧ m(c2 ) ∧ ... ∧ m(cn )|ci ∈ C, n ∈ N} any basic Proof. Take C uniform cover of H(A). Then = {g(t(c1 ∨ c2 ∨ ... ∨ cn ) ∧ m(c1 ) ∧ m(c2 ) ∧ ... ∧ m(cn ))|ci ∈ C, n ∈ N} g(C) = {c1 ∧ c2 ∧ ... ∧ cn |ci ∈ C, n ∈ N}. so g(C) ∈ μ and hence g is uniform. The map g is onto as we saw earlier, and Now C ≤ g(C), ≤ C, so g is also a surjection. for any C ∈ μ we have g(C) Proposition 4.14. For any frame map f : A → B between frames, there exists a frame homomorphism f˜ from H(A) to H(B) such that f˜(m(a)) = m(f (a)) and f˜(t(a)) = t(f (a)) for each a ∈ A, and making the following diagram commutative: H(A)
f˜
gA
H(B) gB
f
A
B
Proof. Let XA be the set consisting of all symbols t(a) and m(a) as a varies over A, and likewise for the set XB . Let F XA and F XB be the free frames respectively, and let νA and νB be the corresponding quotient maps defined earlier. Consider the diagram XA
iA
F XA
νA
F XA /ΦA = H(A)
p
XB
f˜
ψ iB
F XB
νB
H(B)
where p is the map m(a) → m(f (a)), t(a) → t(f (a)), and ψ is the unique frame homomorphism such that ψ ◦ iA = iB ◦ p. Thus ψ(m(a)) = m(f (a) and ψ(t(a)) = t(f (a) for all a ∈ A. It can be verified that the relation RA on F XA that determines H(A) is such that RA ⊆ ker(νB ◦ ψ). Thus there exists a unique frame homomorphism f˜: H(A) → H(B) such that f˜ ◦ νA = νB ◦ ψ. If we suppress the quotient maps and think of the m(a) and the t(a) as elements of the Vietoris frame, then we get f˜(m(a)) = m(f (a)) and f˜(t(a)) = t(f (a)) for each a ∈ A. Finally, for any a ∈ A we have (gB ◦ f˜)(m(a)) = gB (f˜(m(a))) = gB (m(f (a)) = f (a), and (f ◦ gA )(m(a)) = f (gA (m(a))) = f (a). A similar calculation works for the t(a). Thus gB ◦ f˜ agrees with f ◦ gA on the generators of H(A) and hence they are equal.
THE VIETORIS UNIFORMITY FOR LOCALES
9
Proposition 4.15. If A and B are uniform frames and f : A → B is a uniform frame map, then so is f˜: H(A) → H(B). Proof. Take any basic uniform cover C˜ of H(A), where C is a uniform cover of A. Now f (C) is ˜ which then implies f˜(C) ˜ is a uniform a uniform cover of B. We show that in fact f (C) = f˜(C), cover. For this take any t(f (c1 ) ∨ ... ∨ f (cn )) ∧ m(f (c1 )) ∧ ... ∧ m(f (cn )) ∈ f (C, and note that t(f (c1 ) ∨ ... ∨ f (cn )) ∧ m(f (c1 )) ∧ ... ∧ m(f (cn )) = t(f (c1 ∨ ... ∨ cn )) ∧ m(f (c1 )) ∧ ... ∧ m(f (cn )) = f˜(t(c1 ∨ ... ∨ cn )) ∧ f˜(m(c1 )) ∧ ... ∧ f˜(m(cn )) = f˜(t(c1 ∨ ... ∨ cn ) ∧ m(c1 ) ∧ ... ∧ m(cn )). Thus f˜ is uniform.
Observe that if we have a diagram f
h
→B− →C A− ˜ ◦ f˜: For any a ∈ A we have in UniFrm then h ◦f =h ˜ ◦ f˜)(m(a)) = h(m(f ˜ (h (a)) = m(h(f (a)) = (h ◦ f )(m(a)), ˜ ◦ f˜. and the same for the t(a). Since they agree on the generators of H(A) we have h ◦f = h is the identity on H(A). Thus Similarly one can see that if id : A → A is the identity, then id we have: Theorem 4.16. The assignment of a uniform frame A to its Vietoris uniform frame H(A), and of a uniform frame map f : A → B to the uniform map f˜: H(A) → H(B) makes H : UniFrm → UniFrm a functor. 4.3. Completion. Theorem 4.17. If A is a uniform frame such that H(A) is complete, then A must be complete. Proof. As we saw earlier, A is embedded as a closed uniform sublocale of H(A). Since closed uniform sublocales of complete uniform locales are complete in the induced uniformity ([1]), the result follows. In the classical theory of hyperspaces, Morita ([8]) showed that if a uniform space is complete then its hyperspace of non-empty compact subsets with its hyperspace uniformity must be complete. We think that the result is also true in the pointfree setting, but we have not been able to prove this. We conclude with this open question. Question. If a uniform frame A is complete, is it true that H(A) is complete? Acknowledgements. We are grateful to Aleˇs Pultr for the many valuable discussions relating to this work. We also thank the referee for helpful comments and suggestions which allowed us to improve the presentation of our paper. References [1] B. Banaschewski, Uniform completion in pointfree topology, in: Topological and Algebraic Structures in Fuzzy Sets, pp. 19-56. Trends Log. Stud. Log. Libr., vol. 20, Kluwer Academic Publishers, Boston, Dordrecht, London, 2003. [2] B. Banaschewski and A. Pultr, Samuel compactification and completion of uniform frames, Math. Proc. Camb. Phil. Soc. 108 (1990), 63-78. [3] A. Illanes and S. B. Nadler, Hyperspaces, Marcel Dekker (1999). [4] J. R. Isbell, Uniform Spaces, Amer. Math. Soc., Providence (1964). [5] P. T. Johnstone, Stone Spaces, Cambridge University Press (1982).
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DHARMANAND BABOOLAL AND PARANJOTHI PILLAY
[6] P. T. Johnstone, Vietoris Locales and Localic Semilattices, Continuous lattices and their applications (Bremen, 1982), 155-180, Lecture Notes in Pure and Appl. Math., 101, Dekker, New York (1985). [7] E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182. [8] K. Morita, Completion of hyperspaces of compact subsets and topological completion of open-closed maps, General Topology and Appl. 4 (1974), 217-233. [9] J. Picado and A. Pultr, Frames and Locales, Springer, Basel (2012). School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa Email address: [email protected], [email protected]