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The Locally Fine Coreflection and Normal Covers in the Products of Partition-complete Spaces

We prove that the countable product of supercomplete spaces having a countable closed cover consisting of partition-complete subspaces is supercomplete with respect to its metric-fine coreflection. Thus, countable products of sigma-partition-complete paracompact spaces are again paracompact. On the other hand, we show that in arbitrary products of partition-complete paracompact spaces, all normal covers can be obtained via the locally fine coreflection of the product of fine uniformities.

spaces satisfying a structural recursive condition (C-scattered,Čech-scattered) Noetherian trees were applied to show that paracompactness (among others) is preserved in countable products of such spaces. These results extended the initial theorems of Frolík [7] and Arhangel'skii [3] that paracompactness is preserved in countable products of locally compact spaces, and similar results from [36] (scattered spaces) and [6] (Cscattered spaces), including the result of [1] on C-scattered Lindelöf spaces. However, and more importantly, the new results showed that the open covers of such product could be refined by ones generated recursively from open covers of the factors.
For paracompact products, the above condition on the locally fine coreflection is equivalent to the spatiality of the corresponding product of locales, noted by Isbell in [24]. Spatiality was extensively studied by Plewe in [32], who used the productivity of paracompactness in locales and spatiality to extend ( [33]) the above topological results to countable products of C δ -absolute spaces. These spaces are equivalent to those with a complete exhaustive sieve studied by Michael in [28] and to the partition-complete spaces of Telgársky and Wicke in [42]. Unfortunately, spatiality does not extend to spaces which are merely countable unions (e. g., the rationals) of closed partition-complete parts. As an extreme case, Q × Q is not spatial.
It has been recently shown by the first author ( [16]) that for spaces with pre-uniformities, there is a simple relationship between the 'covering monoid' of their localic product and the locally fine coreflection of the product pre-uniformity. The covers consisting of open rectangles of the latter product form a monoid which naturally maps onto a 'generating monoid' of the localic product. It follows that for the products of regular spaces with supercomplete pre-uniformities, the locally fine coreflection of the product pre-uniformity is supercomplete provided that the product is spatial. (We call a pre-uniformity supercomplete if it contains every open cover of the underlying space.) On the other hand, the locally fine pre-uniformities are equivalent to the so-called formal spaces, for which the essential questions about covers are related to their effective computability. Thus, we come back to our methods of recursive constructions by means of Noetherian trees.
As we cannot rely on the spatiality of the products, we will construct the refinements directly. Moreover, we will handle arbitrary (in general non-paracompact) products of partition-complete regular spaces and show that their locally fine covers by regular open sets belong to (i. e., are refined by members of) the locally fine coreflection of any supercomplete uniformities of the factors.
Before starting our preliminary section for necessary definitions, we state our two main results: Theorem A (Theorem 6.4): Let (X i : i ∈ N) be a countable family of σ-partition-complete, regular spaces, and let µ i be a supercomplete pre-uniformity for X i . Then λmΠµ i is a supercomplete pre-uniformity for the topological product ΠX i .
Theorem B (Theorem 7.5): Let (X i : i ∈ I) be a family of partition-complete, completely regular spaces, and for each i ∈ I, let µ i be a supercomplete uniformity for X i . Then every normal cover V of X = Π I X i belongs to λΠ I µ i .

Preliminaries.
Here we take up definitions of several tools needed in the main part of this paper. The locally fine coreflections of uniformities were introduced in [9], and they were constructed internally by means of consecutive 'derivatives' closing the given (pre-) uniformity under the combinatorial condition of local fineness. In the theory of 'formal spaces', this closure is considered via the condition of composition ( [5]) or transitivity ( [37]) of the related covering relation. It may be regarded as an attempt to obtain the underlying topology of the space through iterated applications of uniform subdivisions of covers. We recall the definition of λ from [11] by means of slowed-down Ginsburg-Isbell derivatives. Let µ, ν be pre-uniformities on a set X. Then ( [34]) µ/ν denotes the pre-uniformity generated by all covers of X of the form : α < β} for a limit ordinal β. The first µ (α) unchanged under this derivation is denoted by λµ and called the locally fine coreflection of µ. It is the minimal pre-uniformity closed under the operation ν → ν/ν finer than µ.
If V ∈ λµ, we call V a λ-cover (with respect to µ) or directly a λµ-uniform cover. For a subset A ⊂ X, a λ-neighbourhood (with respect to a given pre-uniformity µ) is a set N ⊃ A for which there is V ∈ λµ such that St(A, V) ⊂ N . We note here that if the µ i are pre-uniformities and V ∈ λΠµ i , then V has a refinement W ∈ λΠµ i consisting of basic rectangles, i. e., sets of the form where F is a finite set. This result can be easily proved by using the inductive definition of the operation λ.
Noetherian trees. Well-founded trees in the context of locally fine refinements were first used in [31] and in an early version of [30]. They were consequently applied in several papers: [18], [19], [20], and in [17] in the context of well-founded cubical triangulations refining open covers of infinite-dimensional cubes.
In this paper, a tree is a set T with a partial order ≤ such that T has a unique minimal element (root) with respect to ≤ and for each p ∈ T , the set of ≤-predecessors of p is linearly ordered and finite. We call a tree Noetherian if each of its linearly ordered parts is finite. To fix some notation, we denote by S(p) the set of all immediate successors of p ∈ T . The symbol End(T ) denotes the set of all maximal elements of T .
Thus, if T is Noetherian, then not only is End(T ) non-empty, but each maximal linear part (branch) of T meets End(T ) in a unique element.
The Ginsburg-Isbell derivatives are related with Noetherian trees as follows. Let µ be a pre-uniformity on a set X. We say that a mapping ϕ : T → 2 X satisfies the uniform mapping condition with respect to µ if for each p ∈ T , the family {ϕ(q) : q ∈ S(p)} is a µ-uniform cover of ϕ(p), i. e., there is a cover U ∈ µ such that U ↾ ϕ(p) ≺ ϕ(S(p)), or in still other words, if ϕ(S(p)) ∈ µ ↾ ϕ(p). The principle introduced by the third author in [30] states that U ∈ λµ if, and only if, there is a Noetherian tree T and a mapping ϕ : T → 2 X , satisfying the uniform mapping condition with respect to µ, such that ϕ(End(T )) ≺ U. We may delimit our discussion to trees consisting of subsets of X, and define the uniform refinement condition with respect to the associated inclusion of the tree in 2 X . Perverse products. We will define special 'weak' subproducts of direct products of trees with the property that (under suitable conditions) infinite chains project onto infinite chains in the factor trees. In this paper, a perversity is a decreasing sequence p : N → Z which is eventually zero. (This concept is modified from that used in intersection homology theory, cf. [27].) Sets of perversities, ordered by the coordinatewise partial order of functions N → Z, are used to reduce a given direct product of countably many trees. Let (T i : i ∈ N) be a countable family of trees, and let P be a set of perversities.
The perverse product Π P T i is the subset of the direct product Π i∈I T i consisting of all elements x for which there is a perversity p ∈ P such that for each i ∈ N, the level of x i in T i , i. e., the number of predecessors in the tree, equals p(i). In addition to countable products, we define the finite perverse products in the same way, by considering the restriction P ↾ F = {p ↾ F : p ∈ P} of perversities to the given finite set F ⊂ N.
The set-theoretic direct product of the T i has as its elements the Cartesian products Π i p i (where each factor p i is considered a set), and the set-theoretic perverse product is the corresponding subproduct.
The perverse product of countably many trees is a tree, provided that the associated family of perversities is itself a tree under its natural partial order. The perverse product of trees is a partially ordered set such that infinite chains project onto infinite chains in the factors, provided that the set of perversities has the same property.
We now define a standard set P = {p n : n ∈ N} of perversities to be used in the sequel. Let p 1 = (0, 0, 0, . . .). Suppose that p n = (i 1 , . . . , i m , 0, 0, . . .) has been defined. Put Then define p n+1 = (i 1 , . . . , i k + 1, i k+1 , . . . , i m , 0, 0, . . .). The elements of this sequence are (), (1), (1, 1), (2, 1), (2, 1, 1), (2, 2, 1) etc., where we have indicated the non-zero entries only. The set P is linearly ordered with respect to the coordinatewise order, and hence any perverse product of trees with respect to P is again a tree. To give an example of its use, let T be a tree, and let T ′ = (Π i∈N T ) P be the perverse set-theoretic power of T with respect to P as explained above. The root of T ′ (the unique element of level 0) is the Cartesian power R N of the root of T , and the elements of level 3, say, are products where P 1 is of level 2 and P 2 is of level 1 in T .
K-scattered spaces and K-derivatives. Let K be a class of topological spaces. A space X is called K-scattered [40], [41] if every non-empty closed subset of X contains a point having a K-neighbourhood in this subset. We define the K-derivatives of a K-scattered space X as follows. Let D K (X) denote the subset of all p ∈ X having no K-neighbourhood. Then D K (X) is a closed subset of X, and we set D K (X)), and for a limit ordinal β set D (β) K (X). The K-rank of X is defined as the smallest ordinal α for which D α K (X) = ∅. Thus, the K-rank of a locally K space is at most one.
Decomposition trees. We will assume that K is hereditary with respect to closed subsets. The Kdecomposition tree T K (X) of X will be defined as follows. The elements of T K (X) are closed subsets of X defined by using the derivative sets D (α) K (X). Letᾱ = Rank K (X). Thus, Dᾱ K (X) = ∅, but D α K (X) = ∅ for all α <ᾱ. The closed subset {D α K (X) : α <ᾱ} is denoted by the symbol top K (X); notice that this is a locally K subset such that every point p of X − top K (X) has a closed neighbourhoodŪ p such that Rank K (Ū p ) <ᾱ. Also note thatᾱ is a limit ordinal if, and only if, top K (X) is empty. We let Root(T K (X)) = X, and the set of immediate successors of X is defined to consist of the set top K (X) and of all such closed subsetsŪ of X − top K (X) with non-empty interior and for which Rank K (Ū ) <ᾱ. If Rank K (X) = 1, then T K (X) = {X}; otherwise, the setsŪ satisfy Rank K (Ū ) <ᾱ, and we can (recursively) define the tree T K (X) by hanging top K (X) and the trees T K (Ū ) below X. Then T K (X) is a well-founded tree.
3. Partition-completeness and stationary winning strategies. this section, we consider a condition that generalizes both C-scattered andČech-complete spaces. Recall that a sequence U = (U n ) of covers of a space X is complete if any filter base F 'controlled' by U has a cluster point, in other words if any filter F with F ∩ U n = ∅ for all n satisfies {F : F ∈ F } = ∅. The classical characterization from Frolík [8] and Arhangel'skii [2] describes theČech-complete spaces as the completely regular spaces with a complete sequence of open covers.
On the other hand, K-scattered spaces have canonical exhaustions into left-open partitions of subsets the well-order of P n+1 is compatible with that of P n in the sense that the relation Q ⊂ P between elements P ∈ P n , Q ∈ P n+1 preserves the order, and 3) any filter base controlled by P has a cluster point. We call a space σ-partition-complete if it is the union of countably many partition-complete, closed subspaces.
The results important for us were already obtained by E. Michael [28]. Recall that a space is K-scattered if every non-empty closed subset of the space contains a point with a K-neighbourhood in the subset. Michael We will use this condition in terms of a topological game G(X) related to such sequences of covers.
Players I and II choose in alternating steps non-empty subsets S 1 ⊃ T 1 ⊃ S 2 ⊃ · · · of X such that Player Notice the connection between K-scattered spaces and stationary winning strategies in the above sense.
By providing for each non-empty closed subset F ⊂ X an open set ∅ = φ(F ) ⊂ F , the winning strategy φ yields a decreasing decomposition of X into 'derivative' subsets and thus (inverting the order) into a Noetherian decomposition tree. The φ-derivative of a non-empty subset S ⊂ X is simply defined as the relatively closed set S \ φ(S). The consecutive derivatives are then defined in a complete analogy with the definition given in the previous section. We denote the corresponding decomposition tree by T φ (X).
If X is partition-complete with respect to a winning strategy φ, we extend its decomposition tree T φ (X) to a (in general) non-Noetherian 'game tree' as follows. For each U ∈ End(T φ (X)), let A(U ) be the collection of all closed F ⊂ U of the formV , where V ⊂ U is non-empty and open. In order to ensure that U is the union of such closures, we assume that X is regular. Denote by φ S the restriction of φ to the subsets of S, for any subset S ⊂ X. Extend T 1 = T φ (X) to T 2 by declaring theV ∈ A(U ) as the immediate successors of U , and hanging the trees T φV (V ) belowV . This operation is repeated countably many times; we obtain an increasing sequence T 1 , T 2 , . . . of Noetherian trees, and define T (X) as their union. Although T (X) in general is not well-founded, it has the following Noetherian property with respect to open covers. If G is an open cover of X, then there is a Noetherian subtree T ′ of T (X) such that End(T ′ ) ≺ G <ω , where G <ω denotes the directed open cover obtained as the finite unions of elements in G. This follows from the following general (and simple) lemma: Lemma 3.3: Let X be a topological space, and let F = (F n : n ∈ N) be a decreasing sequence of closed subsets such that any filter finer than F has a cluster point. Then {F n : n ∈ N} is compact.
4. The λ-neighbourhood induction lemma. Our main method for proving that a given cover G belongs to the locally fine coreflection applies the following 'Noetherian' induction: There is a Noetherian tree T such that each maximal element E of T has a λ-neighbourhood U with the property that G ↾ U is a λ-cover of U . Then the construction of T as a (subtree of) perverse power and the induction lemma given below enable us to conclude the validity of this property for the top element of T . This was stated and proved in [20] for uniform spaces, but it is valid for pre-uniformities (covering monoids).
Lemma 4.1: Let X be a set, and let µ be a pre-uniformity on X.
5. The Metric-fine Coreflection. We call a cover G of a uniform space µX σ-uniform if there is a countable closed cover {F n : n ∈ N} of X such that each restriction G ↾ F n is a uniform cover of the subspace F n . This property can be formulated in several other settings. One calls [10] a uniform space µX metricfine, if for any metric uniform space and any uniformly continuous map f : µX → ρM , f remains uniformly continuous when the target ρM is changed to F M , i. e., the fine space associated with M . The metric-fine coreflection mµ is the weakest metric-fine uniformity stronger than µ. We note that mµ is is the collection of all σ-uniform covers of X. Each σ-uniform (open) cover has a σ-uniformly discrete (open) refinement. A cover G is σ-uniform if, and only if, there is a sequence (U n : n ∈ N) of uniform covers of µX such that for (Notice the relation with the so-called θ-refinability of (topological) spaces). The metric-fine coreflection was used in both [19] and [20] to extend the results from K-scattered spaces to σ − K-scattered, where K was the class of compact (resp.Čech-scattered paracompact) spaces. The metric-fine coreflection m can be directly extended to pre-uniformities of topological spaces.
In our situation, we define the metric-fine modification mµ of a pre-uniformity µ as the filter of all covers refined by covers V for which there is a countable closed cover {F n : n ∈ N} of the underlying space such that for each n ∈ N, there is U n ∈ µ with U n ↾ F n ≺ V ↾ F n .
The covers in λmµ are obtained by means of Noetherian trees as follows. There is a Noetherian tree T and mapping ϕ : T → 2 X with the property that for each p ∈ T , the immediate successors S p of p in T are divided into countably many parts S (n) p such that for each n, the image ϕ(S (n) p ) is a uniform cover of the closed subspace (ϕ(S (n) p )) of ϕ(p), and these subspaces form a countable cover of ϕ(p).

Countable Products of σ-partition-complete supercomplete spaces.
In our proof of Theorem 7.5 that the locally fine coreflection of the product λΠF X i contains all the normal covers, we will need the result that each finite product is supercomplete †. We establish here a stronger result for countable products. Indeed, we don't have to stop at paracompactness. We will prove a product theorem for regular spaces X having supercomplete pre-uniformities (monoids of covers), i. e., filters µ of coverings such that each open cover of X is in µ. We first state the result for partition-complete spaces.
Theorem 6.1: Let (X i : i ∈ N) be a countable family of partition-complete, regular spaces, and for each i, let µ i be a supercomplete pre-uniformity for X i . Then λΠµ i X i is a supercomplete pre-uniformity for the topological product ΠX i .
Proof. The proof is very similar to the one given in [20] for the case of countably manyČech-scattered paracompacta. Therefore, we will only give the main steps of the proof and refer the reader to [20] for details. Let G be an open cover of ΠX i . We may assume that the factors X i are the same space X. (Replace the factors by their disjoint union.) For each i, let T be the tree T φ (X), where φ is the stationary winning strategy. Let T ′ be the perverse power Π P T , where P is the standard set of perversities defined above (Section 2). Finally, let T ′′ be the subtree of T ′ consisting of all P ∈ T ′ which do not have a predecessor Q with a uniform neighbourhood N such that G is a uniform cover of N . (The term 'uniform' refers here to the related pre-uniformities.) We claim that T ′′ is well-founded. Indeed, suppose to the contrary that T ′′ has an infinite chain (P n : n ∈ N). By the definition of perverse products, the projections (π i [P n ] : n ∈ N) are again sequences with infinitely many distinct terms. Thus there are, for each i, relatively open sets E . Therefore, the intersection is compact, and it follows that It is easy to find (finite) families B i of open subsets of the spaces X i which cover the sets K i such that for some s ∈ N, it is not sufficient that all the finite products are supercomplete. It was shown in [14] that there is a subspace X of the real numbers such that each finite power (FX) n is supercomplete but (FX) ω is not. The space X is not partition-complete. On the other hand, it was shown in [15] that there are non-analytic subspaces X of R such that λ((FX) κ ) = (FX) κ for all κ.
of K, and therefore so is G. Now for each i ∈ {1, . . . , s} there is n i such that π i [P ni ] ⊂ N i . Put n = max{n 1 , . . . , n s }. Then it is easy to see that P n ⊂ N , which yields a contradiction, because P n+1 is an element of T ′′ .
We prove by using the λ-neighbourhood induction lemma that G is a λ-uniform cover of the root of T ′′ .
Let P be an element of T ′′ . If P is a maximal element of T ′′ , then by the definition of T ′′ , there is a uniform neighbourhood N P of P such that G is a uniform cover of N P . (More precisely, there is a cover U in Πµ i such that U ↾ N P ≺ G ↾ N P .) We proceed by induction and assume that every successor Q of P in T ′′ has a λ-uniform neighbourhood N Q such that G is a λ-uniform cover of N Q . Since T ′ is a perverse product, and since T ′′ is a lower part (an ideal) of T ′ , there is i ∈ N such that the immediate successors Q of P differ only with respect to i: We recall that T (X) was obtained through a countable iteration of the trees T φ (S) for closed subsets S of X, defining it as the union of the trees T n . There are two possibilities. Either π i [P ] is a maximal element of some T n and its immediate successors all belong to T n+1 , or else both π i [P ] and its immediate successors belong to the same tree of the form T φ (S). However, in each case, the immediate successors are divided into two parts: a closed set F ⊂ π i [P ] and all the closuresV of non-empty open sets V ⊂ π i [P ] \ F such that ∩ P be the immediate successor of P that corresponds to F . By the induction hypothesis, Q F has a λ-uniform neighbourhood N F such that G is a λ-uniform cover of N F . Then P − N F has a closed λ-uniform neighbourhood V F such that V F ∩ Q F = ∅, and such that the binary cover {N F , V F } is a λ-uniform cover of P . Indeed, there is a λ-uniform cover V of X ω such that St(P − N F , V) ∩ Q F = ∅.
(Simply take V as a λ-uniform cover such that St(Q F , V) ⊂ N F .) We can assume that V consists of the closures of basic open sets. This follows from the assumption that the spaces X i are regular and that every open cover of X i belongs to µ i . (Recall that the locally fine coreflection of a product of pre-uniformities has a basis of covers consisting of rectangular elements.) Then each element V of V ↾ P that meets P − N F satisfies π i [V ] ∩ π i [Q F ] = ∅, which implies that V is an immediate successor of P . But then by the inductive hypothesis, each V ∈ V ↾ P with V ∩ (P − N F ) = ∅ has a λ-uniform neighbourhood N V such that G is λ-uniform cover of N V . It then follows that P has a λ-uniform neighbourhood N P satisfying the condition of our claim. In fact, by the λ-neighbourhood induction lemma the set N F has a λ-uniform neighbourhood The sets V such that V ∩ (P \ N F ) = ∅ together with N F form a λ-uniform cover of P . As G is λ-uniform over the sets N V and N ′ F , the set P has by a new application of the lemma a λ-uniform neighbourhood N P with the desired property.
For the following corollaries, notice that the pre-uniformities generated by all locally finite, locally countable, point-finite, point-countable, and countable open covers are locally fine (i. e., stable under the operation λ). Corollary 6.2: Let (X i : i ∈ N) be a countable family of regular paracompact (resp. para-Lindelöf, metacompact, meta-Lindelöf, Lindelöf) spaces. If the spaces X i are partition-complete, then the product space ΠX i is paracompact (resp. para-Lindelöf, metacompact, meta-Lindelöf, Lindelöf).
Proof. Suppose that the spaces X i are paracompact and partition complete. Let µ i be the preuniformity formed by all locally finite open covers of X i . Then µ i is supercomplete for Ω(X i ), because by paracompactness every open cover of X i can be refined by, and thus is, a member of µ i . By the theorem proved above, λΠµ i is supercomplete pre-uniformity for the topology of ΠX i . But Πµ i has a basis consisting of products of finitely many locally finite open covers, and therefore so does λΠµ i , because the pre-uniformity This corollary can be stated also for other locally fine covering concepts, e. g., ultraparacompactness (refinement by clopen covers) etc. It was essentially proved by Plewe in [33]. He showed that the localic product of countably many partition-complete regular spaces is spatial. The result then follows from a theorem of Dowker and Strauss ( [4]) that the localic product of regular paracompact (resp. metacompact, Lindelöf, ultraparacompact) spaces is again paracompact (resp. metacompact, Lindelöf, ultraparacompact).
The case of paracompactness was already due to Isbell [24]. One could also obtain theorem 6.1 by combining Plewe's result on spatiality and the relationship between λΠµ i and ⊗Ω(X i ), mentioned above. In addition to the topological corollaries, theorem 6.1 yields stronger combinatorial corollaries † for uniformities. Corollary 6.3: Let (µ i X i : i ∈ N) be a countable family of supercomplete spaces. If the spaces X i are partition-complete, then the product space Πµ i X i is supercomplete.
Proof. As the spaces µ i X i are supercomplete, we have λµ i = F (X i ) for each i. Therefore, by 6.1, where we have used [22], Exercise VII 8a (p. 143).
As in [19], [20], we may extend the proof of 6.1 to the σ-partition-complete case. (Recall that a space is called σ-partition-complete if it is a countable union of partition-complete, closed subspaces.) Thus, we obtain the following result: Theorem 6.4: Let (X i : i ∈ N) be a countable family of σ-partition-complete, regular spaces, and let µ i be a supercomplete pre-uniformity for X i . Then λmΠµ i is a supercomplete pre-uniformity for the topological product ΠX i .
Proof. Given a family (X i : i ∈ N) of regular spaces with supercomplete pre-uniformities µ i , each of which is a countable union of closed, partition-complete subspaces, we replace the trees T φ (X i ) with a forest which contains a subtree T φ (F n ) for each of the partition complete parts F n . The perverse product construction ensures that only products of partition-complete parts are used. Since the partially ordered set T ′′ so constructed is Noetherian, T ′′ only has countably many 'types' of subproducts. We use the same induction as in 6.1 , but we arrive, instead of the entire product of the X i , at minimal elements P which are subproducts and have (λ-uniform) neighbourhoods N P such that G ω is λ-uniform over N P . As the successors of these minimal elements belong to the same type (i. e., their factors are contained in the same partition-complete parts), and there are only countably many types, we arrive at a countable, closed cover of the product such that G is λ-uniform over each member. Consequently, λ(mΠµ i ) is a supercomplete pre-uniformity. † We also note that it gives a basis for effective product theorems, inasmuch as the factor spaces can be presented effectively.
7. Normal covers in uncountable products. In [30], the third author proved that every normal cover of an arbitrary product of complete metric spaces belongs to the locally fine coreflection of the corresponding product of the fine uniformities of the factor spaces. This result had been previously established for arbitrary products of Polish spaces (cf. [23], VII, Cor. 21.). In this section we will extend this to products of paracompact, partition-complete spaces. It turns out as in the preceding section that we may consider supercomplete pre-uniformities of regular partition-complete spaces.
The proof is based on simple combinatorial properties of 'basic subsets' of products. Let (X i : i ∈ I) We first recall lemmas from [30]. It is useful to begin with the following: Lemma 7.1: Let B 1 , B 2 be basic sets in a product ΠX i , and let F = I(B 1 ) ∩ I(B 2 ). Suppose that In this paper, we will use two extended forms of these lemmas for finite unions of basic sets. They follow from the above versions. Proof. Indeed, this is weaker than the statement for one-element collections. Proof. By the assumption, and by the preceding lemma, Suppose that V is a normal cover, i. e., V is uniformizable. Let V 1 be a normal cover with V 1 ≺ * * V, and let R be a locally finite, open refinement of V 1 (which every normal cover has). Then the regular open extension In the following proof, we will replace the given normal cover V by the regular open extension of R <ω , where R <ω is the directed cover consisting of the finite unions of elements of R, and R is as above. Indeed, let V, V 1 and R be as above. We claim that the regular open extension (R <ω ) * of R <ω refines V <ω . To see this, let R 1 , . . . , R n ∈ R, where we have sets We are now able to state the main result of this section.
Theorem 7.5: Let (X i : i ∈ I) be a family of partition-complete, completely regular spaces, and for each i ∈ I, let µ i be a supercomplete uniformity for X i . Then every normal cover V of X = Π I X i belongs to λΠ I µ i .
Proof. Let R be a locally finite cover of X by regular open sets such that (R <ω ) * ≺ V <ω , and let G be a cover consisting of basic open sets such that each member of G meets only finitely many members from R. We will show that V <ω belongs to λΠ I µ i . We conclude that every normal cover belongs to this product uniformity.
We may assume that the cover R is non-trivial, i. e., R = X for all R ∈ R. Given any G ∈ G, consider the finite set R G ⊂ R such that G ∩ R = ∅ for all R ∈ R \ R G . Then B ∈ B(R) for such an R implies G ∩ B = ∅ and hence by Lemma 7.1, I(G) ∩ I(B) = ∅. On the other hand, as R is non-trivial, Lemma 7.4 implies that for each R ∈ R G there is a finite set F (R) ⊂ I which meets each I(B), B ∈ B(R). Hence, is a finite set which satisfies F ∩ I(B) = ∅ for all B ∈ B(R) and all R ∈ R. Recall from 6.1 that for the associated finite product, λΠ F µ i contains every open cover of Π F X i .

Consider the open cover
of the subproduct Π F X i of X. (Here B(R) denotes the family of all basic sets contained in some member of R.) By the result just mentioned, W F belongs to λ(Π F µ i ). Hence, there is a Noetherian tree T F consisting of subsets of Π F X i satisfying the uniform refinement condition with respect to Π F µ i and for which End(T F ) ≺ W F . Moreover, we may assume that the elements of T F are closures of basic open sets.
We extend T F to a Noetherian tree T ′ F by adding, for each end element P , the perverse product of the trees T φi (P i ) (with respect to the index set F ) below P , where P = ∩{π −1 i [P i ] : i ∈ F }. (We order the subset F as {x 1 , . . . , x n } and use the the same perversities as in 6.1 . The finite enlargements F (P ) defined below will extend this order. Notice that this limits the level of the elements from T φi (P i ) to n.) Finally, we extend T ′ F to a tree T 0 in P(Π I X i ) by crossing each element with Π I\F X i . Then this new tree satisfies the uniform refinement condition with respect to Π I µ i .
Let End * T 0 be the set of all end elements P of T 0 for which there is no R ′ ∈ (R <ω ) * , say R ′ = (R 1 ∪ · · · ∪ R n ) * , with open basic subsets B ∈ B(R) such that B i ⊂ R i and P ⊂ B 1 ∪ · · · ∪ B n . If End * T 0 = ∅, we continue the inductive definition of T . Let P ∈ End * T 0 . Then π F [P ] refines, by the definition of T F , the cover π F [G]; let G ∈ G be such that π F [G] ⊃ π F [P ]. On the other hand, π F [P ] also refines π F [W]; let , where B is a basic subset and B ⊂ R for some R ∈ R.
Denote by R(P ) the set of all finite subsets R ′ ⊂ R, say R ′ = {R 1 , . . . , R n }, for which there are and additionally B i ∩ P = ∅ for all i ∈ {1, . . . , n}. Thus, R(P ) = ∅, established by the one-element subsets. The set R G of all R ∈ R which meet G is finite. Consider an arbitrary element R ′ ∈ R(P ), R ′ = {R 1 , . . . , R n }, and a corresponding set {B 1 , . . . , B n } of basic sets B i ∈ B(R i ) as defined above. In case On the other hand, given R ′ ⊂ R G , Lemma 7.3' implies that there is a finite set F (R ′ ) ⊂ I such that wheneverB and I(B) are as above. (Because otherwise we would have P ⊂ (∪R ′ ) * , which is ruled out by the assumption P ∈ End * (T 0 ).) Put and note that F (P ) is finite. Let As before, there is a Noetherian tree T P consisting of subsets of P satisfying the uniform refinement condition with respect to (Π F (P ) µ i ) ↾ P and such that End(T P ) ≺ W P . For the following and subsequent perverse products, we order the set F (P ) ⊃ F extending the order of F . Extend T P to a Noetherian tree T ′ P by adding the perverse product of the trees T φi (π i [Q]) above the elements Q, where Q ∈ End(T P ) and i ∈ F (P ). As above, we enlarge T ′ P to a Noetherian tree in P(Π I X i ) by crossing the elements of T ′ P with the product of the X i for all the remaining factors. When this is done for each P ∈ End * (T 0 ), we obtain a Noetherian tree T 1 .
In general, if n > 0 and T n has been constructed, then the inductive step from T n to T n+1 is entirely analogous to the above construction: We consider the set End * (T n ) of all end elements P for which there is no finite subset R ′ ⊂ R with P ⊂ (∪R ′ ) * . Given such an element Q, there is a unique P ∈ End * (T n−1 ) below Q, and we define R(Q) as the set of all finite subsets R ′ = {R 1 , . . . , R n } ⊂ R for which there are basic open sets B i ⊂ R i such that π F (P ) [B 1 ∪ · · · ∪ B n ] ⊃ π F (P ) [Q] and B i ∩ Q = ∅ for all i. The construction of T ′ Q then proceeds as above. We set Finally, we put T = ∪{T n : n ∈ N}.
We claim that T is Noetherian. Indeed, suppose to the contrary that T has an infinite chain (Q n : n ∈ N).
Now each T n is Noetherian, so there is in fact an infinite chain (P n : n ∈ N) of elements P n ∈ End * (T n ) such that P k ⊃ Q n(k) . We have P n+1 ⊂ P n and F (P n ) ⊂ F (P n+1 ). Let J = ∪{F (P n ) : n ∈ N}. The ordering of T along the chain (P n ) is, because of the perverse orderings of the added trees T ′ P , an ordering for which there is an infinite sequence (π i [P n(i) ]) of subsets of X i for all i ∈ J such that π i [P (n+1)(i) ] ⊂ π i [P n(i) ]. By the construction there are relatively open sets U n,i = φ i (π i [P n(i) ]) such that π i [P (n+1)(i) ] ⊂ U n,i ⊂ π i [P n(i) ], where φ i denotes the stationary winning strategy associated with the space X i .

It follows that each
is non-empty and compact. As I(P n ) ⊂ J for all n, and each P n is basic, we obtain ∩ n∈N P n = π J [∩ n∈N P n ] × Π I\J X i = ∅ and the projection image (denote it by K) is compact. Each x ∈ ∩P n has a basic open set B x such that x ∈ B x ⊂ R x for some R x ∈ R. There is a minimal finite set E such that In fact, there is n such that k ≥ n implies π J [P k ] ⊂ ∪{π J [B x ] : x ∈ E}.
(To see this, assume that each π J [P k ] meets the complement of ∪{π J [B x ] : x ∈ E}, say, in a closed set A k . Then for each i ∈ J, the sets π i [A k ] form a filter base finer than (U n,i ), and thus have cluster points x i ∈ K i . But K is the product of the K i and the point x = (x i ) i∈J of K has a neighbourhood which does not meet any member of the filter base (A k ), contradicting that x is a cluster point of the latter.) Formal spaces essentially are counterparts of locally fine pre-uniformities in pre-orders (P, ≤) (partial order without antisymmetry), defined by 'covering relations' µ ⊂ P × 2 P which satisfy the following axioms: 1) If a ∈ U , then µ(a, U ). a ≤ b, then µ(a, {b}).

2) If
3) If µ(a, U ) and µ(a, V ), then µ(a, U ∧ V ), where U ∧ V denotes the set of all b ∈ P majorized by both U and V . 4) If µ(a, U ) and for all u ∈ U , µ(u, V ), then µ(a, V ).
A set U ∈ 2 P such that (a, U ) ∈ µ is considered a 'cover' of the element a ∈ P . By the above axioms, the covers of a form a 'locally fine' filter for each a ∈ P . (For a pre-uniformity µ, one takes a filter of covers of the maximal element (the underlying set)).
Their 'effective constructions' are connected -via the above theorem -with the constructions for locally fine coreflections of products. However, we leave the details of obtaining effective refinements of open covers of products to future papers. Let it be mentioned, however, that the definition of the product of covering relations is reflected in the perverse product of trees. In the application of perverse products in Section 6, the immediate successors of a given element vary with respect to one coordinate only. Furthermore, the set of perversities is given effectively, and so is the associated product, relative to the factors.