On covariant functors in the category of compact Hausdorff spaces

Topology and its Applications 179 (2015) 111–121

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On covariant functors in the category of compact Hausdorff spaces A.V. Ivanov ∗ , K.V. Matyushichev a r t i c l e

i n f o

Article history: Received 30 January 2014 Accepted 15 May 2014 Available online 8 September 2014 Dedicated to the memory of prof. V.V. Fedorchuk

a b s t r a c t We investigate covariant functors in the category of compact Hausdorff spaces and obtain a series of results allowing to construct normal and seminormal functors of finite degree with desired properties. In particular, it turns out that any functor of finite degree is a factor-functor of some monomorphic functor of the same degree. Under CH, a compact discretely generated space Z is constructed such that F (Z) is not discretely generated for any seminormal functor F of finite degree. © 2014 Elsevier B.V. All rights reserved.

MSC: 54B15 54B30 54B35 54C05 54C10 54C15 54C60 54D30 Keywords: Seminormal functor Functor of finite degree The Basmanov mapping Discretely generated space

1. Introduction The notion of a normal functor in the category of compact Hausdorff spaces was first introduced by E.V. Ščepin in [15]. A classical result of Katětov says that if the 3rd power of a compact Hausdorff space is hereditarily normal, then the space is metrizable. In 1987 V.V. Fedorchuk [8] generalized the Katětov theorem to any normal functor of degree ≥ 3. Later, the Fedorchuk theorem was transferred to a wide class of seminormal functors. Finally, under the assumption of CH all seminormal functors of finite degree, for which the generalized Katětov theorem holds, were described in [11]. A similar situation arises when one tries to transfer the A.V. Arhangel’ski˘ı and A.P. Kombarov theorem [1] (which says that if the space X 2 \ Δ is normal (here X is an arbitrary compact Hausdorff space and Δ = {(x, x) : x ∈ X}), then χ(X) ≤ ω0 ) * Corresponding author. http://dx.doi.org/10.1016/j.topol.2014.08.021 0166-8641/© 2014 Elsevier B.V. All rights reserved.

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to normal and seminormal functors other than that of raising to the second power. It turns out that under the assumption of Jensen’s Principle (♦) this theorem does not hold for functors expn (see [13]). At the same time, the A.V. Arhangel’ski˘ı and A.P. Kombarov theorem can be generalized to the so called regular functors. Such results require rather subtle investigations of the structure of the functors of finite degree. This article is mainly devoted to the study of properties of such functors. Let us mention some of the results we obtained. It is proved that a functor of finite degree is seminormal if and only if this functor is finitely monomorphic, preserves a one-point space, preserves the empty space, and preserves an empty intersection of finite spaces. It is proved that the Basmanov mapping πFXn (see [4]) is continuous for any (not necessarily monomorphic) functor of finite degree. On the basis of the Basmanov theorem (see [3]), normality and seminormality of functors of degree n are characterized in terms of their action in the category n. This makes it possible to construct functors of finite degree with desired properties. It is proved that any (not necessarily monomorphic) functor of degree n is a factor functor of some monomorphic functor of the same degree. Some of the results are concerned with interrelations between the properties of finite monomorphicity, preservation of finite intersections, and continuity of functors. The last section of the article is devoted to the question whether seminormal functors of finite degree preserve the property of being discretely generated. Originally, this question arose (see [5]) for the functor of raising to the second power: is it true for a discretely generated compact Hausdorff space Y that Y 2 is also discretely generated? A counterexample (denoted by X) to the question was constructed in [14] under the assumption of CH. It is worthwhile to note that in constructing X the method of resolvents in the sense of V.V. Fedorchuk (see [7]) is applied. Thus, in the terminology of [2] the space X is an F -compact space (a Fedorchuk compact space). It turns out that the space X can serve as a starting point in constructing a discretely generated compact space Z such that F(Z) is not discretely generated for any seminormal functor F of finite degree. 2. Definitions and some properties of functors We do not distinguish between ⊂ and ⊆. Comp denotes the category of all compact Hausdorff spaces and their (continuous) mappings. Only full subcategories of the category Comp and only covariant functors between them will be considered. By Comp∗ we denote Comp without the empty space. For any X, Y ∈ Comp let Y X be the set of all the continuous mappings f : X → Y . For any f ∈ Y X let Im(f ) = f (X) ⊂ Y be the image of X. Further, Comp0 = {X ∈ Comp : dim(X) = 0}. If A ⊂ X then iA,X denotes the inclusion mapping A → X which sends each point of A to itself (in X). A functor F is monomorphic [15] if for every embedding i : Y → X the mapping F(i) : F(Y ) → F(X) is an embedding, too. For a monomorphic functor F and a closed subset A ⊂ X the space F(A) is naturally identified with the subspace F(iA,X )(F(A)) of the space F(X). Observe that any functor maps homeomorphisms to homeomorphisms. Remark 2.1. Let A be a closed subset of a compact space X, for which there exists a retraction r : X → A. Then r ◦ iA,X = idA . Consequently, the mapping F(iA,X ) is an embedding and F(A) can be considered as a subspace of F(X). One readily observes that the mapping F(r) : F(X) → F(A) is a retraction, too. Let us note that all the above statements hold for every finite subset A of any zero-dimensional compact space X. They say that a functor F preserves intersections [15] if for any X and any family {Aα } of closed subsets Aα ⊂ X the following equality holds:

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F

    {Aα } = F(Aα ) .

113

(2.1)

We will say (see Basmanov [3]) that a functor F preserves non-empty intersections if equality (2.1) holds  as long as the intersection {Aα } is non-empty. A functor F is continuous [15] if for any inverse spectrum S = {Xα , pα β } the space F(lim S) is naturally homeomorphic to the limit of the spectrum F(S) = {F(Xα ), F(pα )}. β A continuous monomorphic functor F that preserves intersections is said to be seminormal [9], if F preserves a one-point space and the empty space, i.e. |F(X)| = |X| if |X| = 0, 1. A functor F is called epimorphic if for any surjective mapping f : X → Y the mapping F(f ) is also surjective. They say that a monomorphic functor F preserves inverse images if for any f : X → Y and any closed A ⊂ Y the equality F(f )−1 (F(A)) = F(f −1 (A)) holds. A functor F preserves weight if for any infinite compact space X the equality w(X) = w(F(X)) holds. A seminormal functor is called normal if it is epimorphic, preserves inverse images, and preserves weight. Classically, functors of finite degree were defined only along with their monomorphicity (see [3,9]). In the rest of the article we will need the following generalized definition of functors of finite degree. Definition 2.2. We will say that a functor F has a degree deg F ≤ n if for any X F(X) =



 F(iA,X ) F(A) : A ⊂ X, |A| ≤ n .

(2.2)

(For a monomorphic functor in (2.2) F(iA,X )(F(A)) may be replaced by F(A), and then we arrive at a classical definition of a functor of degree ≤ n.) Definition 2.3. A functor F is finitely monomorphic if for any X and any finite A ⊂ X the mapping F(iA,X ) is an embedding. If F is a finitely monomorphic functor and A is a finite subset of X then F(A) can be regarded as a subspace of F(X). By virtue of Remark 2.1 any functor F : Comp0 → Comp is finitely monomorphic. Proposition 2.4. A finitely monomorphic functor F of finite degree is monomorphic. Proof. Let deg F ≤ n, Y ⊂ X, ξ1 , ξ2 ∈ F(Y ), ξ1 = ξ2 . There exist subsets A1 , A2 ⊂ Y such that |Ai | ≤ n and ξi ∈ F(Ai ) ⊂ F(Y ), i = 1, 2. Let A = A1 ∪ A2 . Then ξ1 , ξ2 ∈ F(A) ⊂ F(Y ). We have iY,X ◦ iA,Y = iA,X and F(iA,X )(ξ1 ) = F(iA,X )(ξ2 ). Consequently, F(iY,X )(ξ1 ) = F(iY,X )(ξ2 ), as required. 2 Definition 2.5. We will say that a finitely monomorphic functor F preserves the empty intersection of finite sets if for any X and any disjoint finite A, B ⊂ X the equality F(A) ∩ F(B) = ∅ holds. Theorem 2.6. A functor F of finite degree is seminormal if an only if it is finitely monomorphic, preserves the empty set, preserves a one-point set, and preserves the empty intersection of finite sets. Proof. Necessity is obvious. Let us prove sufficiency. By virtue of Proposition 2.4 F is monomorphic. A theorem of Basmanov (see [3], Theorem 1.3) states that any monomorphic functor of finite degree preserves non-empty intersections. If a family {Aα } of closed subsets of a compact space X has an empty intersection k−1 k k then there is a subfamily Aα1 , . . . , Aαk such that B = i=1 Aαi = ∅ and i=1 Aαi = ∅. If i=1 F(Aαi ) = ∅ then there exists ξ ∈ F(B) ∩ F(Aαk ) = ∅. Take finite subsets C1 ⊂ B and C2 ⊂ Aαk such that ξ ∈ F(Ci ), i = 1, 2. But this is impossible, because C1 ∩ C2 = ∅ and the functor F preserves the empty intersection of finite sets. Thus, F preserves intersections. A theorem of E.V. Ščepin (see [15], Proposition 3.7) says that any monomorphic functor of finite degree is continuous. 2

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It is well known that a monomorphic functor of finite degree is epimorphic (see [3]). Proposition 2.7. Any functor F of finite degree is epimorphic. Proof. Let deg F ≤ n, f : X → Y be surjective and ξ ∈ F(Y ). There exist A ⊂ Y and η ∈ F(A) such that |A| ≤ n and F(iA,Y )(η) = ξ. Take a mapping g : A → X such that f ◦g = iA,Y . Then F(f )(F(g)(η)) = ξ. 2 Example 2.8. The functor κ. Let X be a compact space and x ∈ X. By Cx we denote the component of x. Consider the equivalence relation R on X defined by the formula xRy ⇔ Cx = Cy . Let κ(X) = X/R (a point in κ(X) is a component in X). Since for any x ∈ X the component Cx is the intersection of all clopen subsets of X containing x (see [6, Theorem 6.1.23]), the space κ(X) is a zero-dimensional compact Hausdorff space. For any f : X → Y define κ(f )(Cx ) = Cf (x) . This determines the functor κ : Comp → Comp0 and for any X ∈ Comp0 we have κ(X) = X. It is easy to see that deg κ = 1 and the functor κ is not finitely monomorphic. This example shows that the Basmanov theorem mentioned in the proof of Theorem 2.6 (i.e., every monomorphic functor of finite degree preserves non-empty intersections) cannot be generalized to arbitrary functors of finite degree (with the obvious replacement of F(Aα) ⊂ F(X) by F(iAα ,X )(F(Aα ))). Indeed, let X be a compact space containing only two components C1 , C2 and x ∈ C1 , y, z ∈ C2 , y = z. Define A1 = {x, y}, A2 = {x, z}, A = A1 ∩ A2 . Then





{C1 } = κ(iA,X ) κ(A) = κ(iA1 ,X ) κ(A1 ) ∩ κ(iA2 ,X ) κ(A2 ) = {C1 , C2 }. Let F be a monomorphic functor preserving intersections. Then for any X and any ξ ∈ F(X) the support of ξ is defined: supp ξ =



 A : A ⊂ X, ξ ∈ F(A) .

Evidently, ξ ∈ F(supp ξ) and supp ξ is the smallest closed subset of X with this property. For any mapping f : X → Y and any ξ ∈ F(X) there is an inclusion

supp F(f )(ξ) ⊂ f (supp ξ),

(2.3)

because f ◦ isupp ξ,X = if (supp ξ),Y ◦ f |supp ξ and, as a consequence, F(f )(ξ) ∈ F(f (supp ξ)) (see [9]). For any n, any X, and any monomorphic functor F preserving intersections define   Fn (X) = ξ ∈ F(X) : | supp ξ| ≤ n ⊂ F(X), the closed subspace of F(X) (see [10, Proposition 1.2]). For any f : X → Y let Fn (f ) = F(f )|Fn (X) . In this way for any n we obtain a subfunctor Fn of the functor F (see [10, Proposition 1.5]). Observe that the definition of subsets Fn (X) given above can be generalized to arbitrary functors F: Fn (X) =





 F(iA,X ) F(A) : A ⊂ X, |A| ≤ n .

But the following example shows that subspaces Fn (X) may not be compact even for a monomorphic functor F preserving finite intersections, which means that equality (2.1) holds for any finite family {Aα }. Example 2.9. The functor βd . For any compact space X by Xd denote the set X equipped with the discrete topology. Let βd (X) = β(Xd ), where β is the Čech–Stone compactification. Every mapping f : X → Y

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is continuous with respect to the discrete topologies on X and Y . There is a unique extension βd (f ) of the mapping f : Xd → Yd over the Čech–Stone compactifications β(Xd ) and β(Yd ) (see [6]). In this way we obtain a functor βd : Comp → Comp. Properties of the Čech–Stone compactification imply that βd is monomorphic and preserves finite intersections. At the same time (βd)1 (X) = Xd ⊂ β(Xd ), i.e. for any infinite X (βd )1 (X) is not closed in βd (X) and is not a compact space. In the sequel, by a natural number n we will also denote the discrete space n = {0, . . . , n − 1} consisting of n points. For any functor F, any X, and any n the Basmanov mapping [4] πn = πFXn : X n × F(n) → F(X), is defined by the formula πn (a, ξ) = F(a)(ξ), where a ∈ X n , ξ ∈ F(n) and a is considered as a mapping a : n → X. If a functor F is continuous then the mapping πn is continuous for any X (see [4]). From the definition it follows immediately that Im πn ⊂ Fn (X). For a monomorphic functor F preserving intersections we have Im πn = Fn (X) ([10, Proposition 1.3]). Theorem 2.10. For any functor F of finite degree, any n, and any X ∈ Comp the mapping πFXn is continuous. Proof. The functor F|Comp0 is finitely monomorphic and of finite degree. Consequently, F|Comp0 is monomorphic. Then, by virtue of a theorem of E.V. Ščepin (see [15]) F is continuous in the category Comp0 , i.e. commutes with the limit of an inverse spectrum consisting of zero-dimensional compact spaces. Hence, the mapping πFXn is continuous for X ∈ Comp0 (see [4]). Now, let X be an arbitrary compact space and f : Y → X a mapping of some zero-dimensional compact space Y onto X. The following diagram commutes Y n × F(n)

πF Y n

f n ×idF (n)

X n × F(n)π

F(Y ) F(f )

F Xn

F(X)

For any subset U ⊂ F(X) we have n

 −1

−1 πF πFY n F(f )−1 U Xn (U ) = f × idF(n) −1 (here for any g : Z → T and any A ⊂ Z by g  (A) we denote the set {t ∈ T : g −1 t ⊂ A}). Hence, πFXn (U ) is open for any open U ⊂ F(X). 2

Remark 2.11. It is easy to see that if F is a functor of finite degree and deg F ≤ n, then the mapping πFXn is surjective for any X. It can be shown that for any functor F of finite degree and any spectrum S = {Xα , pα β } the mapping lim F(pα ) : F(lim S) → lim F(S) is surjective (here pα : lim S → Xα denote the limit projections of the spectrum S). Question 2.12. Is it true that any functor of finite degree is continuous? Let F be an arbitrary functor. As in [10], for any n we set Fnn (X) = Fn (X)\Fn−1 (X) and F0 (X) = F(∅).

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For a functor F the set   sp(F) = k : k ∈ N, Fkk (k) = ∅ . is said to be its degree spectrum (see [11]). 3. The Basmanov theorem on definition of monomorphic functors of finite degree by their action on the category n Following Basmanov [3], by n we also denote the category consisting of only one object n and all the mappings of n into itself. The following theorem holds. Theorem 3.1. (Basmanov [3]) Any functor F : n → Comp can be canonically extended to a monomorphic functor F B : Comp → Comp that preserves intersections and is of degree ≤ n. Moreover, if G : Comp → Comp is a monomorphic functor preserving intersections of degree ≤ n and n > 1, then (G|n )B = G, where G|n is the restriction of G to the category n. The idea of constructing the functor F B consists in the following (see [3]). The action of the functor F on the mappings of the category n allows to introduce an equivalence relation R on the product X n × F(n). Namely, (a, ξ), (b, η) ∈ X n × F(n) are said to be comparable if there exists f : n → n such that either a = b ◦ f , η = F(f )(ξ) or b = a ◦ f , ξ = F(f )(η). Points t, s ∈ X n × F(n) are said to be equivalent if there exists a finite sequence t1 , . . . , tk where all neighbors are comparable and t1 = t, tk = s. The quotient space X n × F(n)/R induced by this relation is defined as F B (X). Moreover, the natural quotient mapping of X n × F(n) onto F B (X) turns out to be the Basmanov mapping for the functor F B . In what follows we shall call the functor F B the Basmanov extension of a functor F : n → Comp. As a corollary, from the Basmanov theorem we deduce that for any monomorphic functor G : Comp → Comp that preserves intersections the equality (G|n )B = G|n , n > 1 holds. Let us note that for n = 1 the second part of the Basmanov theorem does not hold. Consider two functors: F 1 (X) = X × [0, 1] and F 2 (X) = Con(X) (the last one is the result of collapsing the top X × {1} of the cylinder X × [0, 1] to the point {1}; in particular, Con(∅) = {1}). Both these functors are monomorphic, preserve intersections, are of degree 1 and, nevertheless, F 1 |1 = F 2 |1 but F 1 = F 2 . Let F : n → Comp and F(n) = ∅. Further, let fi be a constant mapping of n into n, for which Im fi = {i}, i ∈ n. We shall say that F preserves constant mappings if the mapping F(fi ) : F(n) → F(n) is also constant (it is clear that a particular i ∈ n does not play any role here). We shall say that F distinguishes between constant mappings if F(fi ) = F(fj ) as soon as i = j (i, j ∈ n). Theorem 3.2. Let F : n → Comp. The functor F B is seminormal if and only if F preserves constant mappings and distinguishes between them. Proof. Necessity. Let fi : n → n be a constant mapping. Then, by virtue of (2.3) F(fi )(F(n)) ⊂ F B (fi (n)). So, fi (n) = {i} and F B preserves a one-point space. Therefore, |F B (fi (n))| = 1. Given i = j, we have







F(fi ) F(n) ∩ F(fj ) F(n) ⊂ F B fi (n) ∩ F B fj (n)

= F B fi (n) ∩ fj (n) = F B (∅) = ∅. Hence, F(fi ) = F(fj ). Sufficiency. The functor F B is monomorphic, preserves intersections, and is of degree ≤ n. Therefore, F B is continuous. Represent a mapping fi as a composition fi = i{i},n ◦ f¯i , where f¯i : n → {i}. Since F B (i{i},n ) is

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an embedding, the set F B (f¯i )(F(n)) = F B ({i}) is a singleton. Hence, the functor F B preserves a one-point set. If i = j (i, j ∈ n) then



∅ = Im F(fi ) ∩ Im F(fj ) = F B {i} ∩ F B {j} = F B (∅). Thus, F B preserves the empty space. 2 For any seminormal functor F : Comp → Comp a space X is naturally identified with F1 (X) ⊂ F(X) (a point x ∈ X corresponds to a unique point of the set F({x}) ⊂ F(X) (see [10])). Thus, for any seminormal functor F and any X we have X ⊂ F(X). In particular, if a functor F : n → Comp satisfies the conditions of Theorem 3.2, then n ⊂ F(n) and every point i ∈ n is identified with a unique point of the set Im F(fi ). Example 3.3. The functors θn . Let θn (n) = n ∪ {ξi : i ∈ n} (ξi = ξj for distinct i, j). Suppose f ∈ nn . Define θn (f ) in the following way. For any f and any i ∈ n let θn (f )(i) = f (i). If f is a bijection, then θn (f )(ξi ) = ξf (i) . If f is not a bijection, then θn (f )(ξi ) = f (i). In this way we obtain a functor on the category n, which preserves constant mappings and distinguishes between them. The Basmanov extension of this functor to the category Comp is the required functor θn . Observe that θ2 (X) = X 2 . One may readily show that sp θn = {1, n}. In [12] (Proposition 2.6.10) one can find a definition of a union F ∨ G (denoted by F  G in [12]) of two seminormal functors F and G. For any X the space (F ∨ G)(X) is defined as the result of gluing together F(X) and G(X) along X. The action of the union of two functors on mappings is defined in a natural way. In exactly the same way one can define a union of n seminormal functors n 

F i.

i=1

One can readily check that the union of seminormal functors is a seminormal functor and sp

n  i=1

F

i

=

n 



sp F i .

i=1

By means of the union of functors it is easy to construct a seminormal functor the degree spectrum of which is equal to any prescribed finite subset of N as long as it contains 1. Indeed, if {1, n1 , . . . , nk } ⊂ N , then k

 ni sp θ = {1, n1 , . . . , nk }. i=1

Let F : n → Comp and F B be its corresponding seminormal functor. Then for every ξ ∈ F(n) the support supp ξ ⊂ n is defined. If A ⊂ n and f : n → n is a retraction of n onto A such that F(f )(ξ) = ξ,

(3.1)

then by virtue of Remark 2.1 we have ξ ∈ F B (A) and supp ξ ⊂ A. At the same time for any retraction f ∈ nn of the space n onto supp ξ equality (3.1) holds. Thus, the support supp ξ can be defined in terms of the functor F as the smallest subset A ⊂ n, for which any retraction f of n onto A satisfies condition (3.1). We shall say that F preserves supports if for any ξ ∈ F(n) and any f ∈ nn the following equality holds: f (supp ξ) = supp F(f )(ξ).

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Theorem 3.4. Let F : n → Comp. The functor F B is normal if and only if F preserves constant mappings and distinguishes between them, preserves supports, and w(F(n)) ≤ ω0 . Proof. Necessity follows from the fact that for the seminormal functor F B the property of preserving inverse images is equivalent to the condition of preserving supports (see [10, p. 130]). Sufficiency. The functor F B is seminormal (Theorem 3.2) and epimorphic (Proposition 2.7). Let X be an infinite compact space and πn : X n × F(n) → F B (X) the Basmanov mapping. Then, Im πn = F B (X) and w(F B (X)) ≤ w(X n × F(n)) = w(X), from which we obtain w(F B (X)) = w(X), because X ⊂ F B (X). It remains to verify the condition of preserving supports. Let f : X → Y , ξ ∈ F B (X) and η = F B (f )(ξ). Then

F B (f |supp ξ )(ξ) = η ∈ F B f (supp ξ) . Pick up embeddings g : supp ξ → n, h : f (supp ξ) → n and a mapping s : n → n such that s ◦g = h ◦f |supp ξ . Let ξ  = F B (g)(ξ), η  = F B (h)(η). Then, F(s)(ξ  ) = η  and, consequently, s(supp ξ  ) = supp η  . At the same time, s(supp ξ  ) = s(g(supp ξ)) = h(f (supp ξ)). Hence, supp η  = h(f (supp ξ)), from which it follows that supp η = f (supp ξ). 2 Remark 3.5. From Theorem 3.4 it follows that the functors θn constructed above satisfy all the conditions of normality, except for the property of preserving inverse images. Theorem 3.6. Let F be an arbitrary functor of degree n. Then there exist a monomorphic functor G of the same degree and a natural transformation j : G → F such that for every X ∈ Comp the mapping jX is surjective (so, the functor F is a factor-functor of a monomorphic functor of the same degree). Moreover, if X ∈ Comp0 , then jX is a homeomorphism. Proof. By letting G = (F|n )B we obtain the desired functor. For every X ∈ Comp∗ the mapping jX is uniquely defined by commutativity of the diagram (here ρX is the natural quotient mapping and πF Xn (a, ξ) = F(a)(ξ)) X n × F(n)

πF Xn

ρX

F(X) jX

G(X) By virtue of Theorem 2.10 and Remark 2.11 the Basmanov mapping πFXn is continuous and surjective, and so is jX . If necessary, redefine F(∅) by letting F(∅) = G(∅). For any X, Y ∈ Comp∗ and f : X → Y the diagrams πF Xn

X n × F(n) f n ×idF (n)

Y n × F(n)π and

F(X) F(f )

FY n

F(Y )

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X n × F(n)

ρX

f n ×idF (n)

Y n × F(n)

119

G(X) G(f )

ρY

G(Y )

commute, from which we have that j : G → F is a natural transformation with the desired properties. The last statement is an immediate consequence of the fact that the functor F|Comp0 is monomorphic and of finite degree (see Theorem 2.10) and of Theorem 1.7 in Basmanov’s paper [3]. 2 Question 3.7. What is the structure of all those functors of degree n whose action on the category n is the same? In other words, what is mutual relationship of functors F and G of degree n if F|n = G|n ? 4. On finite monomorphicity and the property of preserving finite intersections As noted above, any functor F : Comp0 → Comp is finitely monomorphic. The following proposition was proved by E.V. Ščepin ([15, Proposition 3.2]). Proposition 4.1. If F : Comp0 → Comp is continuous, then F is monomorphic. The following example shows that a finitely monomorphic functor may not be monomorphic. Example 4.2. The functor βC . Let X be a compact space. Let us strengthen the topology of X by declaring  all the components of X to be open. The space thus obtained will be denoted by XC : XC = {C : C is a component of X}. If f : X → Y is a continuous mapping, then f : XC → YC is also a continuous mapping. Let βC (X) = β(XC ), where β is the Čech–Stone compactification. The mapping βC (f ) is defined as the extension of f : XC → YC over the Čech–Stone compactifications of XC and YC . In this way we obtain a functor βC : Comp → Comp. If dim X = 0, then βC (X) = βd (X). If a compact space X consists of finitely many components (in particular, X itself is finite), then βC (X) = X. Hence, the functor βC is finitely monomorphic. At the same time βC is not monomorphic. Indeed, let X be the segment [0, 1] and A a countable closed subset of X. Then, AC is a countable discrete space and XC = X. Further, βC (A) = βN , βC (X) = X and the mapping βC (iA,X ) cannot be an embedding, because 2c = |βN | > |X| = c. The functor βC (like βd ) is not continuous. If S = {Xn , pnk } is a countable inverse spectrum consisting of finite spaces with the countable limit X, then lim βC (S) = lim S = X, but βC (X) = βN = X. The degree spectrum of βC and βd is equal to {1}, deg βC = deg βd = ∞. Question 4.3. Is it true that every finitely monomorphic continuous functor F : Comp → Comp is monomorphic? Question 4.4. Is it true that every functor F : Comp0 → Comp is monomorphic? In [9] it was noted that if a functor F : Comp → Comp preserves finite intersections and is continuous, then F preserves any intersections. At the same time the property of preserving intersections by a monomorphic functor implies the closedness of subspaces Fn (X) ⊂ F(X), n ∈ N . Example 4.5. The functor β exp∞ . Let exp X be the space of all non-empty closed subsets of a compact ∞ space X equipped with the Vietoris topology (see [9]), exp∞ X = n=1 expn X ⊂ exp X, and β exp∞ X the Čech–Stone compactification of the space exp∞ X. Observe that for any X the subspace exp∞ X is dense

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in exp X. For f : X → Y we let exp∞ f = exp f |exp∞ X : exp∞ X → exp∞ Y , where exp f : exp X → exp Y is the exponential mapping corresponding to f (see [9]). Let β exp∞ f be the extension of exp∞ f over the Čech–Stone compactifications of exp∞ X and exp∞ Y . It is easy to check that this completes the definition of the functor β exp∞ : Comp → Comp. Let A be a closed subset of X. The functor exp is monomorphic. Hence, exp(iA,X ) is an embedding. Thus, the mapping exp∞ (iA,X ) also is an embedding and Im(exp∞ (iA,X )) = exp∞ X ∩ exp A is a closed subset of exp∞ X. Consequently, β exp∞ (iA,X ) : β exp∞ (A) → β exp∞ (X) is an embedding. Thus, β exp∞ is a monomorphic functor. The functor exp preserves intersections. Hence, for any finite family A1 , . . . , Ak exp∞



  Ai = exp∞ (Ai ),

  from which we have β exp∞ ( Ai ) = β exp∞ (Ai ). Thus, β exp∞ preserves finite intersections. One readily checks that (β exp∞ )n (X) = expn (X) and, consequently, (β exp∞ )n (X) is closed in β exp∞ (X) for any X. Let us show that the functor β exp∞ may not preserve infinite intersections. Let X = [0, 1] × [0, 1] be ∞ the unit square, A = [0, 1] × {0}, An = [0, 1] × [0, 1/n], n ∈ N , i=1 An = A. Let {ξn : n ∈ N } be a sequence of points in exp X such that ξn ∈ exp∞ An \ exp A and lim ξn = A ∈ exp X. The sets exp∞ A and M = {ξn : n ∈ N } are closed and disjoint in exp∞ X. We have

βM ∩ β exp∞ (An ) = Cl β exp∞ (X) M ∩ exp∞ (An ) = Cl β exp∞ (X) (M ) \ {ξk : k < n} = βM \ {ξk : k < n}. ∞ Hence, n=1 β exp∞ (An ) ⊃ βM \ M = ∅. At the same time, β exp∞ A ∩ βM = ∅. Thus, β exp∞ A = ∞ n=1 β exp∞ (An ), i.e. the functor β exp∞ does not preserve infinite intersections. Therefore we have constructed a monomorphic functor preserving finite intersections, for which the spaces (β exp∞ )n (X) are closed in β exp∞ X, and yet this functor does not preserve infinite intersections. Question 4.6. Is there a monomorphic functor in the category Comp which preserves intersections and is not continuous? 5. Functors and the property of being discretely generated A space X is called discretely generated if the closure of any subset A ⊂ X coincides with its d-closure. By the d-closure of A we understand the union of all the discrete subsets of A (see [5]). Theorem 5.1 (under CH). There exists a discretely generated compact space Z such that for any seminormal functor F, the degree spectrum of which contains n > 1, the space F(Z) is not discretely generated. Proof. In [14] under CH a discretely generated compact space X was constructed, the square of which is not discretely generated. Let X1 , X2 , X3 be three copies of X and Z = X1 ⊕ X2 ⊕ X3 . We shall show that the space Z satisfies the conditions of our theorem. Evidently, Z is discretely generated. Let n ∈ sp F (n > 1) and ξ ∈ F(n), supp ξ = n. Choose distinct points x3 , . . . , xn in X3 ⊂ Z n − 2. The subset H = X1 × X2 × {x3 } × . . . × {xn } × {ξ} ⊂ Z n × F(n) is homeomorphic to X 2 . As a consequence, H is not discretely generated.

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Let πn : Z n × F(n) → F(Z) be the Basmanov mapping. If h1 , h2 ∈ H and h1 = h2 , then supp πn (h1 ) = supp πn (h2 ); hence, πn (h1 ) = πn (h2 ). It means that the mapping πn |H : H → F(Z) is an embedding. Accordingly, the space F(Z) is not discretely generated. 2 References [1] A.V. Arhangel’ski˘ı, A.P. Kombarov, On ∇-normal spaces, Topol. Appl. 35 (1990) 121–126. [2] M.A. Baranova, A.V. Ivanov, On the spectral height of F -normal spaces, Sib. Math. J. 54 (3) (2013) 388–392. [3] V.N. Basmanov, Covariant functors of finite degree on the category of Hausdorff compact spaces, Fundam. Appl. Math. 2 (3) (1996) 637–654 (in Russian). [4] V.N. Basmanov, Covariant functors, retracts and dimension, Dokl. Akad. Nauk SSSR 271 (5) (1983) 1033–1036. [5] A. Dow, M.G. Tkachenko, V.V. Tkachuk, R.G. Wilson, Topologies generated by discrete subspaces, Glas. Mat. 37 (57) (2002) 187–210. [6] R. Engelking, General Topology, Warszawa, 1977. [7] V.V. Fedorchuk, Fully closed mappings and their applications, J. Math. Sci. 136 (5) (2006) 4201–4292. [8] V.V. Fedorchuk, On the Katětov cube theorem, Mosc. Univ. Math. Bull. 44 (4) (1989) 102–106. [9] V.V. Fedorchuk, V.V. Filippov, General Topology, Basic Constructions, Moscow Univ. Press, Moscow, 1988 (in Russian). [10] V. Fedorchuk, S. Todorčević, Cellularity of covariant functors, Topol. Appl. 76 (1997) 125–150. [11] A.V. Ivanov, The Katětov property for seminormal functors, Sib. Math. J. 51 (4) (2010) 778–784. [12] A. Teleiko, M. Zarichnyi, Categorical Topology of Compact Hausdorff Spaces, Mathematical Studies Monograph Series, vol. 5, VNTL Publishers, 1999, 263 pages. [13] A.V. Ivanov, E.V. Kashuba, K.V. Matyushichev, E.V. Stepanova, On seminormal functors and compact spaces of uncountable character, Topol. Appl. 160 (13) (2013) 1606–1610. [14] A.V. Ivanov, E.V. Osipov, Degree of discrete generation of compact sets, Math. Notes 87 (3–4) (2010) 367–371. [15] E.V. Shchepin, Functors and uncountable powers of compacta, Russ. Math. Surv. 56 (3) (1981) 1–71.