ASPECTS OF COMPLETE UNIFORM SPREADS IN FRAMES

We give a new notion of a complete uniform spread in terms of a 
relative kind of uniform local connectedness. Properties of this 
type of (uniform) connectedness are discussed. It is also 
shown that our concept of complete spread is equivalent to that 
of Hunt (1982).


Introduction.
The concept of a complete uniform spread together with its completion was introduced into the category Unif of uniform spaces and uniformly continuous functions by Hunt [6] (see also [7]).In [11], we introduced these concepts into the category UniFrm of uniform frames and uniform homomorphisms.
Section 2 is devoted to properties of uniform local connectedness with respect to along.We also show how the Banaschewski-Pultr uniform frame completion CL of a uniform frame L can be used to obtain point-free analogies of some well-known topological results.Section 3 is devoted to complete spreads.Hunt [8] defines a uniform spread f : (X, ᐁ) → (Y , ) to be complete if (X, ᐁ) is a complete uniform space where ᐁ is a spread uniformity generated by f .We give a new definition of a complete uniform spread in terms of the concept of uniform local connectedness with respect to along.It is then shown that such a definition is equivalent to that of Hunt (1982).
We assume familiarity with general knowledge of frames, especially [9] and uniform frames according to [3,4,5].Briefly, a cover of a frame L is a subset A ⊆ L such that A = e, the top element of L. A cover U of L is said to refine a cover V of L if for each x ∈ U there is y ∈ V such that x ≤ y.We write U ≤ V .Given a cover A of L and x ∈ L, we define the A-star of x to be the element A pre-uniformity UL is called a uniformity on L provided that it satisfies the compatibility condition: for each a ∈ L, where a/C = {x ∈ L | Cx ≤ a}.
Our notions of connectedness are those of Baboolal and Banaschewski [2].Given a frame L, we say that an element z ∈ L is connected if whenever z = x ∨ y with x ∧y = 0, then x = 0 or y = 0, and a cover of L is connected provided that its elements are connected.The frame L is connected whenever its top element e is connected, and it is locally connected if for each x ∈ L.An element x ∈ L is a component of an element y ∈ L if x ≤ y is maximally connected.Throughout the paper, we use the shorter x ≤ c y to mean that x is a component of y.The following concept was introduced in [11].
Now, consider a localic spread h : L → M which is onto with L carrying the fine uniformity U F L. (The fine uniformity on a uniform frame L is the uniformity generated by all normal covers, that is, those covers A of L such that A = A 1 in some sequence is a basis for a uniformity on M. Proof.The proof (which is found in [12]) is a consequence of Baboolal [1, Theorem 3.1].
Definition 1.3.The uniformity generated by the basis of the lemma is called the spread uniformity (on M) and h is called a uniform spread.Hereafter, the spread uniformity on M is denoted by U h M.
Recall [2] that a uniform frame L is said to be uniformly locally connected if each cover V of L is refined by a cover U each of whose elements is connected.So, in the definition of a uniform spread, the underlying uniform frames are uniformly locally connected.Moreover, a uniform spread is a uniform homomorphism: for, suppose

Uniform local connectedness with respect to along.
We introduce the notion of uniform local connectedness with respect to along for uniform frames and prove that for dense surjections, this concept coincides with uniform local connectedness introduced by Baboolal in [1].The significance of this concept is given in Section 3 where our approach to complete uniform spreads has a bearing on uniform local connectedness with respect to along.Definition 2.1.Let h : (L, UL) → (M, UM) be a surjective homomorphism.We say that M is uniformly locally connected with respect to L along h if there is a basis Ꮾ for the uniformity UL such that each cover h[B] is connected for B ∈ Ꮾ, where (2.1) Proposition 2.2.Let h : L → M be a surjective homomorphism.If M is uniformly locally connected with respect to L along h, then M is uniformly locally connected.So, in particular, if M is uniformly locally connected with respect to L along h, then L is uniformly locally connected.
Proof.The second statement is a consequence of a result of Baboolal [1].Assume that h : L → M is a surjective homomorphism and that M is uniformly locally connected with respect L along h.Then given A ∈ UM, we have h[B] ≤ A for some B ∈ UL.There is a basis Ꮾ for UL for which h[U ] is connected, for each U ∈ Ꮾ.Moreover, there is a uniform cover must be uniformly locally connected.

Proposition 2.3. Under the hypothesis of Proposition 2.2, if h : L → M is a dense surjection, then M is uniformly locally connected if and only if M is uniformly locally connected with respect to L along h.
Proof.Suppose that M is uniformly locally connected.Then by hypothesis, UM is generated by all C ∈ UM such that C is a connected cover.We claim that the collection is a basis for UL and that h Then there exists a uniform cover In [3], Banaschewski and Pultr described the uniform (frame) completion of a uniform frame L as a certain quotient of the down-set DL of L. (We refer to this completion as the Banaschewski-Pultr uniform frame completion to distinguish it from Isbell's [9] and Kříž's [10].)Using such a completion, we now give point-free versions of some well-known topological results.The notation used in the applications below evolves from the Banaschewski-Pultr uniform frame completion.
The following result (proved in [12]) is a point-free version of the well-known topological fact that in Unif every uniformly continuous function on a dense subspace of a uniform space into a complete uniform space has a unique uniformly continuous extension (see [5]).Proposition 2.4.Let (L, UL) be a complete uniform frame, let h : (L, UL) → (M, UL) be a uniform frame homomorphism, and let ρ : (N, UN) → (M, UM) be a dense surjection.Then there is a unique uniform frame homomorphism g : Corollary 2.5.Given a dense surjection f i : M i → L i from a complete uniform frame to a uniform frame for i = 1, 2, if h : L 1 → L 2 is an isomorphism, then there exists a unique isomorphism g : Proof.By Proposition 2.4, there exist unique uniform homomorphisms g : Since f 1 and f 2 are monic in RegFrm, it follows from the commutativity of the diagrams that g 3. Complete uniform spreads.Hunt [8] defines a uniform spread (X,f ,Z) to be complete if (X, U) is a complete uniform space, where U is the spread uniformity generated by f on X.The uniform spreads (L,h,M) dealt with here are those for which the "base" uniform frame L is complete with respect to some uniformity on L. (Such frames are called frame complete frames [11].) Our notion of a complete spread is in terms of uniform local connectedness with respect to along.We will show that our definition is equivalent to that of Hunt.In [11], we have shown that our notion of completeness for spreads can be used to obtain point-free versions of Hunt's uniform properties spreads.Definition 3.1.An extension of a uniform spread (L,h,M) is a uniform spread (L,g,N) together with a dense surjection f : N → M where M is uniformly locally connected with respect to N along f such that f • g = h.A uniform spread (L,h,M) is said to be complete if, whenever (L,g,N) is a uniform spread and f : N → M is a dense surjection with M uniformly locally connected with respect to N along f such that the triangle commutes, then f is an isomorphism.Proposition 3.2.The completeness of a uniform spread is frame invariant under spread isomorphisms.
Proof.Suppose that (L,h,M) and (H,g,K) are isomorphic uniform spreads with a spread isomorphism (f , k) : (L,h,M) → (H,g,K) with (H,g,K) complete.To see that (L,h,M) is also complete, take a uniform spread (L,s,N) and a dense surjection t : N → M with M uniformly locally connected with respect to N along t such that It is easily shown that t is an isomorphism.
Proof.Since f is dense, the denseness of the desired homomorphism is immediate.To show that UN ⊆ U g N, we need only to show that for each A ∈ UN there exists We claim that T W ≤ f * [T V ]: take y ∈ T W . Then y ≤ c g(w) for some w ∈ W .But then y ≤ g(v) for some v ∈ V .Now we have f (y) ≤ f • g(v) = h(v).Since M is locally connected (being uniformly locally connected [2]), we may (and do) assume that Now recall that for a dense onto frame homomorphism f : L → M with M locally connected with respect to L along f , the right adjoint f * preserves pairwise disjoint joins [11].So, in particular, we have Proof.( ⇒) Suppose that h : (L, UL) → (M, U h M) is complete and let p : (N, UN) → (M, U h M) be a dense surjection.Since L is complete, it follows that h extends to a uniform spread, say g : L → (N, U g N) by the construction in Proposition 2.4-such an extension is precisely the composition and (trivially) p • g = h.
Since p is a dense surjection, it follows (Lemma 3.3) that p : (N, U g N) → (M, U h M) is also a dense surjection.Now because M is uniformly locally connected, it follows that M is uniformly locally connected with respect to N along p (Proposition 2.3).Therefore, since h is complete and since p • g = h with g being a uniform spread, it is implied that p is an isomorphism, thus, M is a complete uniform frame.

. 4 )Proposition 3 . 4 .
Now since the f * (a) are pairwise disjoint and y is connected it follows that y ≤ f * (a), for some a ≤ c h(v)-which proves thatT W ≤ f * [T V ].A uniform spread h : (L, UL) → (M, U h M) is a complete uniform spread if and only if (M, U h M) is a complete uniform frame; that is, h is completein the sense of Definition 3.1 if and only if it is complete in the sense of Hunt.