Applied General Topology c© Universidad Politécnica de Valencia Volume 3, No. 1, 2002 pp. 33–44 On paracompact spaces and projectively inductively closed functors T. F. Zhuraev Abstract. In this paper we introduce a notion of projectively in- ductively closed functor (p.i.c.-functor). We give sufficient conditions for a functor to be a p.i.c.-functor. In particular, any finitary normal functor is a p.i.c.-functor. We prove that every preserving weight p.i.c.- functor of a finite degree preserves the class of stratifiable spaces and the class of paracompact σ-spaces. The same is true (even if we omit a preservation of weight) for paracompact Σ-spaces and paracompact p-spaces. 2000 AMS Classification: 54D18, 54B30. Keywords: stratifiable space, paracompact σ-space, paracompact Σ-space, paracompact p-space, projectively inductively closed functor. 1. Introduction By Tych we denote the category of all Tychonoff spaces and all their contin- uous functions. A Hausdorff compact space is called a compact space or just a compactum. By Comp we denote the full subcategory of Tych, whose objects are compacta. Recall that a covariant functor F : Comp → Comp is said to be normal [17] if it satisfies the following properties: (1) preserves the empty set and singletons, i.e., F(∅) = ∅ and F({1}) = {1}, where {k} (k ≥ 0) denotes the set {0, 1, . . . ,k−1} of nonnegative integers smaller than k. In this notation 0 = {∅}. (2) is monomorphic, i.e., for any (topological) embedding f : A → X, the mapping F(f) : F(A) →F(X) is also an embedding. (3) is epimorphic, i.e., for any surjective mapping f : X → Y , F(f) : F(X) → F(Y ) is also surjective. (4) continuous, i.e., for any inverse spectrum S = {Xα; παβ : α ∈A} of com- pact spaces, the limit f : F(lim S) → limF(S) of the mappings F(πα), 34 T. F. Zhuraev where πa : lim S → Xα are the limiting projections of the spectrum S, is a homeomorphism. (5) preserves intersections, i.e., for any family {Aα α ∈A} of closed subsets of a compact space X, the mapping F(i) : ∩{F(Aα) : α ∈A}→F(X) defined by F(i)(x) = F(iα)(x), where iα : Aα → X is the identity embeddings for all α ∈A, is an embedding. (6) preserves preimages, i.e., for any mapping f : X → Y and an arbitrary closed set A ⊂ Y , we have F(f−1(A)) = (F(f))−1(F(A)). (7) preserves weight, i.e., w(F(X)) = w(X) for any infinite compactum X. In what follows we shall use bigger than normal classes of functors. But any of them shall preserve empty set, intersections and be monomorphic. By exp we denote the well-known hyperspace functor of non-empty closed subsets. This functor takes every (nonempty) compact space X to the set of all its nonempty closed subsets endowed with the (finite) Vietoris topology (see [9]), and a continuous mapping f : X → Y to the mapping exp(f) : exp(X) → exp(Y ), defined by F(f)(A) = A. For a functor F and an element a ∈ F(X), the support of a is defined as intersection of all closed sets A ⊂ X such that a ∈F(A) (recall that we consider only monomorphic functors preserving intersections). This support we denote by suppF(X)(a). When it is clear what functor and space are meant, we denote the support of a merely by supp(a). A. Ch. Chigogidze [7] extended an arbitrary intersection-preserving mono- morphic functor F : Comp → Comp to the category Tych by setting Fβ(X) = {a ∈F(βX) : supp(a) ⊂ X} for any Tychnoff space X. If f : X → Y is a continuous mapping of Tychonoff spaces and βf : βX → βY is the (unique) extension of f over their Stone-Čech compactifications, then F(βf)(F(βX)) ⊂Fβ(X). The last inclusion is a corollary of a trivial fact (1.1) f(supp(a)) ⊃ supp(F(f)(a)). Therefore, we can define the mapping Fβ(f) = F(βf)|X, which makes Fβ into a functor. A. Ch. Chigogidze proved [7] that if a functor F has certain normality property, then Fβ has the same property (modified when necessary). In what follows by a covariant functor F : Tych → Tych we shall mean a functor of type Fβ. For such a functor F and any compact space X the space F(X) is a compact space. For a set A by |A| we denote the cardinality of A. For a subset A of a space X by A X we denote the closure of A in X. On paracompact spaces and projectively inductively closed functors 35 In this paper we introduce a notion of projectively inductively closed functor (p.i.c.-functor). We give sufficient conditions for a functor to be a p.i.c.-functor (Theorem 3.5). In particular, any finitary normal functor is a p.i.c.-functor (Corollary 3.6). We prove that every preserving weight p.i.c.-functor of a finite degree preserves the class of stratifiable spaces and the class of paracompact σ-spaces (Theorem 3.7). The same is true (even if we omit a preservation of weight) for paracompact Σ-spaces and paracompact p-spaces (Theorem 3.8). All spaces are assumed to be Tychonoff, and all mappings, continuous. Any additional information on general topology and covariant functors one can find, for example, in ([8], [9], [17]). 2. Preliminaries In this section we recall some definitions and facts, which will be useful in establishing our main results (see Section 3). Definition 2.1 ([2]). A network for a space X is a collection N of subsets of X such that whenever x ∈ U with U open, there exists F ∈N with x ∈ F ⊂ U. An elementary corollary of this definition is that every base of a space X is a network of X. A family A of subsets of X is said to be σ-locally finite if it is a union of countably many families An which are locally finite in X. Definition 2.2 ([16]). A topological space X is called a σ-space, if it has a σ-locally finite network. Remark 2.3. A rather simple observation of the definition 2.2 shows us that every closed subset of a σ-space is a σ-space. Proposition 2.4 ([11]). Every closed image of a σ-space is a σ-space. A well-known theorem of E. Michael [13] states that every closed image of a paracompact space is a paracompact space. So, from Proposition 2.4 we get Theorem 2.5 ([11]). Every closed image of a paracompact σ-space is a para- compact σ-space. Theorem 2.6 ([11]). A countable product of paracompact σ-spaces is a para- compact σ-space. In 1969 K. Nagami [15] introduced more general class than class of σ-spaces. Definition 2.7. A space X is a Σ-space if there exists a σ-discrete collection N , and a cover c of X by closed countably compact sets such that, whenever C ∈ c and C ⊂ U with open U, then C ⊂ F ⊂ U for some F ∈N . Clearly, from Definitions 2.1 and 2.7 we have. Proposition 2.8. Every perfect preimage of a σ-space is a Σ-space. In par- ticular, every σ-space is a Σ-space. Proposition 2.9. Every closed subspace Y of a Σ-space X is a Σ-space. 36 T. F. Zhuraev Indeed, evidently, that the families N|Y and c|Y , where N and c are from Definition 2.7, satisfy Definition 2.7 for Y . K. Nagami [15] has shown that the class of Σ-spaces is strictly larger than the class of perfect preimages of σ-spaces. On the other hand, the class of perfect preimages of σ-spaces is much larger than the class of σ-spaces. For example, every compact σ-space is metrizable. Paracompact Σ-spaces behave nicely with respect to countable products and perfect images. Proposition 2.10 ([15]). The countable product of paracompact Σ-spaces is a paracompact Σ-space. Proposition 2.11 ([15]). Every perfect image of a paracompact Σ-space is a paracompact Σ-space. The class of paracompact p-spaces in sense of A. V. Arhangel’-skii is a proper subclass of paracompact Σ-spaces. Definition 2.12 ([3]). A space X is called a p-space if there exists a countable family un such that: 1) un consists of open subsets of βX; 2) X ⊂∪un for each n; 3) ∩nst(x,un) ⊂ X for every x ∈ X. Here for a family v of subsets of a space Y by st(y,v) we denote the set ∪{V ∈ v : y ∈ V}. Theorem 2.13 ([3]). The class of paracompact p-spaces coincides with the class of perfect preimages of metrizable spaces. Corollary 2.14. Every paracompact p-space is a perfect preimage of a para- compact σ-space and, consequently, is a paracompact Σ-space. Theorem 2.13 also yields Corollary 2.15 ([3]). Every countable product of paracompact p-spaces is a paracompact p-space. Proposition 2.16 ([3]). Every closed subspace of a paracompact p-space is a paracompact p-space. Theorem 2.17 ([10]). Every perfect image of a paracompact p-space is a para- compact p-space. Let us recall some more notions and facts. Definition 2.18 ([6]). A space X is stratifiable if there is a function G which assigns to each n ∈ ω and closed set H ⊂ X, an open set G(n,H) containing H such that (1) if H ⊂ K, then G(n,H) ⊂ G(n,K); (2) H = ∩nG(n,H). On paracompact spaces and projectively inductively closed functors 37 The class of stratifiable spaces was defined in 1961 by J.Ceder [6]. But he called these spaces by M3-spaces. The latter form was proposed by C.R. Borges [5] in 1966. In the definition of a stratifiable space we can also use the following addi- tional condition: (3) G(n + 1,H) ⊂ G(n,H). Indeed, we define new stratification G′ by G′(n,H) = ∩i≤nG(i,H). The following dual characterization of stratifiable spaces is sometimes useful: X is stratifiable if and only if for each open U ⊂ X and n ∈ ω one can assign an open set Un such that Un ⊂ U, U = ∪nUn and U ⊂ V implies Un ⊂ Vn. To get this characterization from a function G satisfying definition 2.18, let Un = X\G(n,X\U). On the other hand, to get G from Un’s let G(n,X) = X\(X\H)n. From this characterization of stratifiable spaces and Michael’s theorem [14] characterizing a paracompact space by σ-cushioned coverings, we get Theorem 2.19 ([6]). Stratifiable spaces are paracompact. Corollary 2.20 ([5]). Stratifiable spaces are perfectly normal. Indeed, every stratifiable space is normal in view of Theorem 2.19. On the other hand, each closed subset of X is a Gδ-set by Definition 2.18. Theorem 2.21 ([12]). Every stratifiable space is a σ-space. As a corollary of Theorems 2.19 and 2.21 we get Theorem 2.22. Every stratifiable space is a paracompact σ-space. Theorem 2.23 ([5]). Every subspace of a stratifiable space is stratifiable. Theorem 2.24 ([5]). A countable product of stratifiable spaces is stratifiable. Theorem 2.25 ([5]). Stratifiable spaces are preserved by closed mappings. From Theorem 2.25 we get Corollary 2.26. An image of a metrizable space under closed mapping is stratifiable. In particular, every metrizable space is stratifiable. Going back to functors F : Comp → Comp, we, evidently, have (2.1) a ∈F(supp(a)). If a functor F preserves preimages, then F preserves supports [17], i.e. (2.2) f(supp(a) = supp(F(f)(a)). The property (2.2) can be conversed. Proposition 2.27 ([17]). Any monomorphic preserving intersections functor preserves supports if and only if it preserves preimages. 38 T. F. Zhuraev Definition of the functor F and property (2.2) imply that (2.3) f(suppF(X)(a)) = suppFβ(Y ) Fβ(f)(a) for any preimage preserving functor F : Comp → Comp, continuous mapping f : X → Y , and a ∈Fβ(X). Now we recall one construction given by V. N. Basmanov [4]. Let F : Comp → Comp be a functor. By C(X,Y ) we denote the space of all continuous map- pings from X to Y with compact-open topology. In particular, C({k},Y ) is naturally homeomorphic to the k-th power Y k of the space Y ; the homeomorphism takes each mapping ξ : {k}→ Y to the point (ξ(0), . . . ,ξ(k − 1)) ∈ Y k. For a functor F, compact space X, and a positive integer k, V.N. Basmanov [4] defined the mapping πF,X,k : C({k},X) ×F({k}) →F(X) by πF,X,k(ξ,a) = F(ξ)(a) for any ξ ∈ C({k},X) and a ∈F({k}). When it is clear what functor F and what space X are meant, we omit the subscripts F and X and write πX,k or πk instead of πF,X,k. According to Shcepin’s theorem ([17], Theorem 3.1). the mapping F : C(Z,Y ) →F(F(Z),F(Y )) is continuous for any continuous functor F and compact spaces Z and Y . This implies the following assertion. Proposition 2.28 ([4]). If F is a continuous functor, X is a compact space, and k is a positive integer, then the mapping πF,X,k is continuous. Let Fk be a subfunctor of a functor F defined as follows. For a compact space X, Fk(X) is the image of the mapping πF,X,k and for a mapping f : X → Y , Fk(f) is the restriction of F(f) to Fk(X). Denote by f : C({k},X) → (C{k},Y ) the mapping which takes ξ to composition f ◦ ξ. It is easy to see that (2.4) πY,k ◦f × idF({k}) = F(f) ◦πX,k. Therefore, F(f)(Fk(X)) ⊂Fk(Y ). Hence, Fk is a functor. A functor F is called a functor of degree n, if Fn(X) = F(X) for any compact space X, but Fn−1(X) 6= F(X) for some X. The next assertion (Proposition 2.29) is Shcepin’s definition of the functor Fk. But using Basmanov’s definition we should prove it. One can find the proof in [28]. Proposition 2.29. For any continuous functor F and a compact space X, we have Fk(X) = {a ∈F(X) : |supp(a)| ≤ k}. Corollary 2.30. For any compact space X, we have expk(X) = {a ∈ exp(X) : |a| ≤ k}. On paracompact spaces and projectively inductively closed functors 39 The definition of a support and the property (2.1) imply. Proposition 2.31. For a functor F, a compact space X, and a closed subset A of X, F(A) = {a ∈F(X) : supp(a) ⊂ A}. For a Tychonoff space X, a functor F : Comp → Comp, and a positive integer k, we put Fk(X) = πF,βX,k(C({k}),X) ×F({k})) and denote the restriction of πF,βX,k to C({k})×F({k}) by πF,X,k. If f : X → Y is a continuous mapping, then F(βf)(Fk(X)) ⊂Fk(Y ), in view of the equality (2.4) for the mapping βf. Therefore, setting Fk(f) = Fk(βf)|F(X), we obtained a mapping Fk(f) : Fk(X) →Fk(Y ). Thus, we have defined the covariant functor Fk : Tych → Tych, that extends the functor Fk : Comp → Comp to the category Tych. Proposi- tion 2.29 implies the following assertion. Proposition 2.32 ([18]). If F : Comp → Comp is a continuous functor, then Fk : Tych → Tych is a subfunctor of the functor Fβ, and (2.5) Fk(X) = Fβ(X) ∩Fk(βX). Proposition 2.33 ([18]). For a compact space X, a continuous functor F and a positive integer k, the set Fk(X) is closed in F(X). Propositions 2.32 and 2.33 imply Proposition 2.34 ([18]). For a Tychonoff space X, a continuous functor F, and a positive integer k, the set Fk(X) is closed in Fβ(X). Corollary 2.35. For a Tychonoff space X, a continuous functor F, and a positive integer k, the set Fk(X) is closed in Fk+1(X). 3. Projectively inductively closed functors We start recalling that a functor F is said to be finitely open [18], if the set Fk({k + 1}) is open in F({k + 1}) for any positive integer k. The dual for this definition states that F({k + 1})\Fk({k + 1}) is closed in F({k + 1}). Remark 3.1. As an example of a finitely open functor one can take any finitary functor F, i.e., a functor F such that F({k}) is finite for any positive integer k. In particular, the hyperspace functor exp is a finitary and, consequently, a finitely open functor. 40 T. F. Zhuraev Lemma 3.2. For any continuous, preserving preimages functor Fβ, the map- ping πFβ,X,1 is a homeomorphism. Proof. At first we show that πFβ,X,1 is a bijective mapping. In view of (2.3) for any ξ ∈ C({1},X) and a ∈ F({1}) we have F(ξ)(a) = ξ(0), since we consider functors preserving empty set. Since the set {1} consists of one point 0, every mapping ξ : {1}→ X is a monomorphism. But we consider only monomorphic functors. Hence, the mapping F(ξ) is a monomorphism. On the other hand, πFβ,X,1(ξ,a) = F(ξ)(a) = ξ(0). Consequently, π1 is an injective mapping. Furthere, the mapping πFβ,X,k : C({k},X) ×F({k}) →Fk(X) is epimorphic for any positive integer k, in particular, for k = 1 by definition of F(X). Thus, π1 is a bijective mapping. Hence, π1 is a homeomorphism for a compact space X (π1 is continuous in view of Proposition 2.28). If X is a Tychonoff space, then by definition, the mapping πFβ,X,k is a restriction of πF,βX,k to C({k},X)×F({k}). Therefore, the mapping πF,X,1 is a homeomorphism as a restriction of the homeomorphism πF,βX,1 to a subset. The proof is complete. � Definition 3.3. An epimorphism f : X → Y is called inductively closed if there exists a closed subset A of X such that f(A) = Y and f|A is a closed mapping. Definition 3.4. A functor Fβ is said to be projectively inductively closed (p.i.c.) if the mapping πFβ,X,k is inductively closed for any Tychonoff space X and positive integer k. The next theorem gives us sufficient conditions for a functor Fβ to be pro- jectively inductively closed (a p.i.c.-functor). Theorem 3.5. Every continuous, monomorphic, finitely open functor Fβ : Tych → Tych, that preserves empty set, intersections, and preimages is a p.i.c.- functor. Proof. It is necessary to check, that for any Tychonoff space X and positive integer k, the mapping πFβ,X,k : X k ×F({k}) →Fk(X) is inductively closed. We shall prove it by induction on k. If k = 1, the mapping πF,X,1 is inductively closed, since it is a homeomorphism by Lemma 3.2. Assume that our assertion is proved for all integers k ≤ l. Let us prove it for k = l + 1. Fix some point x0 ∈ Xl. Consider the embedding i: Xl → Xl+1 defined as i(x1, . . . ,xl) = (x1, . . . ,xlx0). Define a mapping j : F({l}) → F({l + 1}) by the equality j(a) = F(h)(a), where h: {l} → {l + 1} is an identical embedding, i.e., h(m) = m for any On paracompact spaces and projectively inductively closed functors 41 m ≤ l−1. Since F is a monomorphic functor, the mapping j is an embedding. Hence, we defined the embedding e = i× j : Xl ×F({l}) → Xl+1 ×F({l + 1}). It follows from definitions that (3.1) πFβ,X,l+1 ◦e = πFβ,X,l. From property (3.1) we get, that on the set e(Xl×F({l})) the next equality holds: (3.2) πF,X,l+1 = πFβ,X,l ◦e −1. Since the mapping πF,X,l is inductively closed by an inductive assumption, there exists a closed subset A of Xl×F({l}) such that πF,X,l(A) = Fl(X), and the mapping πF,X,l|A is closed. Since the mapping e−1 is a homeomorphism on the set e(A), equality (3.2) and Corollary 2.35 imply that (3.3) πF,X,l+1|A is a closed mapping. Moreover, it is clear, that (3.4) πF,X,l+1(A) = Fl(X). Now we put Φ = F({l + 1})\Fl({l + 1}). The set Φ is compact, because the functor F is finitely open. Now we define the sets (3.5) Z0 = X l+1 × Φ and (3.6) Z1 = (βX) l+1 × Φ. By fi, i < 2, we denote restrictions of the mapping πβX,l+1 to the sets Zi. Let us show that (3.7) Z0 = f −1 1 (f1(Z0)). To verify this equality, we remark that the functor Fβ preserves monomor- phisms, intersections, and preimages. Hence, it preserves supports (look at (2.2)). Therefore, (3.8) supp(πl+1(ξ,a)) = ξ(supp(a)) for any ξ ∈ C({l + 1},βX) and a ∈ F({l + 1}). But if (ξ,a) ∈ Z1, then supp(a) = {l + 1}. Consequently, (3.9) supp(f1(ξ,a)) = ξ({l + 1}). Hence, (3.10) Z0 = {(ξ,a) ∈ Z1 : ξ({l + 1}) ⊂ X}. Thus, if f1(ξ0,a0) = f1(ξ1,a1) and (ξ0,a0) ∈ Z0, then (ξ1,a1) ∈ Z0. Hence, equality (3.7) is verified. 42 T. F. Zhuraev Compactness of the set Z1 and the equality (3.7) imply that the mapping f0 : Z0 → f0(Z0) is closed. Now we shall verify that (3.11) f0(Z0) = f1(Z1) ∩Fl+1(X). It is sufficient to check the inclusion ⊃. Let f1(ξ,a) ∈ f1(Z1) ∩Fl+1(X). Then X ⊃ supp(f1(ξ,a)) = ξ({l + 1}) by (3.9). Consequently, (ξ,a) ∈ Z0 according to (3.10). Thus, the equality (3.11) is checked. This equality and compactness of f1(Z1) imply that f0(Z0) is closed in Fl+1(X). Hence, the mapping f0 : Z0 →Fl+1(X) is closed. Let B = A0 ∪Z0. Then the mapping πFβ,X,l+1|B : B →Fl+1(X) is closed as a union of closed mappings on two closed subsets A and Z0. To complete the proof, it suffices to check that (3.12) f0(Z0) ⊃Fl+1(X)\Fl(X). But this inclusion is a corollary of (3.11) and an evident inclusion f1(Z1) ⊃Fl+1(X)\Fl(X). The proof is complete. � Corollary 3.6. Every finitary normal functor, in particular the functor expn, is a p.i.c.-functor. Theorem 3.7. Let Fβ be a weight preserving p.i.c.-functor of a finite degree m. Then Fβ preserves the class of stratifiable spaces and the class of paracompact σ-spaces. Proof. First we consider the case of stratifiable spaces. By Theorem 2.24 the space Xm ×Fβ({m}) is stratifiable as a finite product of stratifiable spaces (Fβ({m}) is stratifiable being a metrizable compact space, because Fβ is a weight preserving functor). Since Fβ is a p.i.c.-functor, there exists a closed subset A of Xm ×Fβ({m}) such that πFβ,X,m(A) = Fβ(X) and the mapping πFβ,X,m|A → Fβ(X) is closed. But every subspace of a stratifiable space is stratifiable in view of Theorem 2.23. Hence, A is a stratifiable space. Then the space Fβ(X) is stratifiable like an image of a stratifiable space under closed mapping (look at Theorem 2.25). � The proof of the assertion for paracompact σ-spaces repeats the previous proof. The necessary changings are the following: instead of Theorems 2.24, 2.23, and 2.25, we use Theorem 2.6, Remark 2.3, and Theorem 2.5 respectively. By the same procedure we get Theorem 3.8. Let Fβ be a p.i.c.-functor of a finite degree. Then Fβ preserves the class of paracompact Σ-spaces and the class of paracompact p-spaces. On paracompact spaces and projectively inductively closed functors 43 Proof of this theorem repeats the proof of Theorem 3.7 for stratifiable spaces. The necessary changings are: 1) we don’t need that Fβ(m) is metrizable; 2) instead of Theorems 2.24, 2.23, and 2.25, we use Propositions 2.10, 2.9, and 2.11 in the case of paracompact Σ-spaces; 3) in the case of paracompact p-spaces we use respectively Corollary 2.15, Proposition 2.16, and Theorem 2.17. Corollary 3.6, Theorems 3.7, and 3.8 yield Corollary 3.9. Every normal finitary functor of a finite degree, in particular the functor expm, preserves the class of stratifiable spaces, the class of para- compact σ-spaces, and the class of paracompact Σ-spaces. Remark 3.10. As for paracompact p-spaces, they are preserved by any normal functor Fβ (look at [1]). References [1] Yu. Al’-Kassas, A metrizability and a paracompactness of probability measures spaces, Vestn. Mosk. Univ. Ser. Matem., Mekh. N1 (1993), 14–17. [2] A. V. Arhangel’skii, An addition theorem for the weight of sets lying in bicompacta, Dokl. Akad. Nauk. SSSR. 126 (1959), N2, 239–241. [3] A. V. Arhangel’skii, On a class of spaces containing all metric spaces and all locally bicompact spaces, Matem. Sbornik. 67 (1965), N1, 55–85. [4] V. N. Basmanov, Covariant functors, retracts, and dimension, Dokl. Akad. Nauk SSSR. 271 (1983), N5, 1033–1036. [5] C. J. R. Borges, On stratifiable spaces, Pacif. J. Math. 17 (1966), N1, 1–16. [6] J. G. Ceder, Some generalizations of metric spaces, Pacif. J. Math. 11 (1961), 105–125. [7] A. Ch. Chigogidze, Extension of normal functors, Vestn. Mosc. Univ. Ser. Matem., Mekh. N6 (1984), 40–42. [8] R. Engelking, General Topology, Warszawa, PWN. 1977. [9] V. V. Fedorchuk and V. V. Filippov, General Topology: Basic Constructions, Moscow, Mosc. State Univ. 1988. [10] V. V. Filippov, On a perfect image of a paracompact p-space, Dokl. Akad. Nauk SSSR. 176 (1967), N3, 533–536. [11] G. Gruenhage, Generalized metric spaces, in Handbook of Set-Theoretic Topology. Am- sterdam. 1984. Ch. 10, 423–501. [12] R. W. Heath and R. E. Hodel, Characterization of σ-spaces, Fund. Math. 77 (1973), N3, 271–275. [13] E. Michael, Another note on paracompact spaces, Proc. Amer. Math. Soc. 8 (1957), 822–828. [14] E. Michael, Yet another note on paracompact spaces, Proc. Amer. Math. Soc. 10 (1959), 309–315. [15] K. Nagami, Σ-spaces, Fund. Math. 65 (1969), 169–192. [16] A. Okuyama, Some generalizations of metric spaces, their metrization theorems, and product spaces, Sci. Rep. Tokyo-Kyoiku-Daigaku. Sect.-A. 9 (1968), 236–254. [17] E. V. Shcepin, Functors and uncountable powers of compact spaces, Uspekhi Matem. Nauk. 36 (1981), N3, 3–62. [18] T. F. Zhuraev, On projectively quotient functors, (submitted). Received April 2001 Revised October 2001 44 T. F. Zhuraev T. F. Zhuraev Department of General Topology and Geometry Mechanics and Mathematics Faculty Moscow State University Vorob’evy Gory, Moscow, 119899 Russia