functors and 1-soft mappings

Topology and its Applications 107 (2000) 183–190

Functors and 1-soft mappings✩

Michael V. Smurov

Chair of General Topology and Geometry, Faculty of Mechanics and Mathematics, Moscow State University,Moscow, 119899, Russia

Received 9 November 1998

Abstract

The properties of the mappingsF(f ) :F(X)→ F(Y ), wheref :X→ Y is a 1-soft mapping ofmetric compact spaces andF a functor from a certain class of finitary covariant functors of finitedegree containing expn and SPnG, are examined. A criterion for the existence of a neighborhood ofa point inF(Y ) such that the restriction ofF(f ) to the preimage of this neighborhood is 1-softis given. The obtained results make it possible to find the degree of topological inhomogeneity ofspacesF(X) for nonmetrizableAE(1)-compactaX homogeneous with respect to character. 2000Elsevier Science B.V. All rights reserved.

Keywords:Compact space; Normal functor; 1-soft mapping;AE(1)-compactum

AMS classification: 54C55; 18B30; 54B30

1. Introduction and preliminaries

This paper gives a detailed study of the properties of mappings that are obtained byapplying finitary functors of finite degree to 1-soft mappings of compact metric spaces.Well-known examples of such functors are expn and SPnG. One of the main results inthis work is Theorem 1, which gives a criterion for a pointa ∈ F(Y ) to have a closedneighborhood such that the restriction ofF(f ) :F(X)→ F(Y ) to the preimage of thisneighborhood is a 1-soft mapping. Theorem 1 and the decomposability of nonmetrizableAE(1)-compacta into inverse systems with soft bounding mappings [2] give Theorems 2and 3 that establish some properties of spacesF(X) for nonmetrizableAE(1)-compactaX.Similar results for the hyperspace functor exp and the functorP of probability measureswere obtained in [4,5].

✩ This work was financially supported by Russian Foundation for Basic Research, project no. 97-01-00357.E-mail address:[email protected] (M.V. Smurov).

0166-8641/00/$ – see front matter 2000 Elsevier Science B.V. All rights reserved.PII: S0166-8641(00)00109-7

184 M.V. Smurov / Topology and its Applications 107 (2000) 183–190

Recall that a compact spaceX is called anabsolute extensor in dimensionn (AE(n)-compactum) if, for any n-dimensional compact spaceZ, an arbitrary continuous mappingf :A→X defined on a closed subsetA of Z has a continuous extensionf :Z→X.

We consider covariant functors on the categoryComp of all compact spaces and theircontinuous mappings. Recall several notions that are used in what follows; see [3,1] formore details.

A functor F is called monomorphic if it maps embeddings into embeddings. For amonomorphic functorF and a closed subsetY ⊂X, the spaceF(Y ) is naturally identifiedwith a subspace ofF(X) (the range of the mappingF(i), wherei :Y →X is the identitymapping). A monomorphic functorF preserves intersections ifF(

⋂{Yα}) =⋂{F(Yα)}for any family {Yα} of closed subsets ofX. For a monomorphic functorF preservingintersections, the support suppa of a point a ∈ F(X) is naturally defined as suppa =⋂{Y ⊂X: a ∈ F(Y )}. Note thata ∈ F(suppa).

A functor F is called normal [3] if F is monomorphic, preserves intersections,commutes with the operation of taking an inverse limit, and preserves (i) the empty set,(ii) the singleton, (iii) epimorphisms, (iv) weight, and (v) supports. The last conditionmeans thatf (suppa)= suppF(f )(a) for anyf :X→ Y anda ∈ F(X).

A normal functorF is called a functor of finite degree if there exists a positive integern

such that, for any compact spaceX, the inequality|suppa|6 n holds for alla ∈ F(X).The smallest suchn is called the degree ofF . A normal functorF is called finitary ifFmaps finite spaces into finite ones.

Definition 1. Suppose thatf :X→ Y is a continuous mapping of compact metric spacesand x ∈ X. The mappingf is called 1-soft at x if it is open atx and, for any openneighborhoodU of x, there exists an open neighborhoodV of x such that the setV ∩ f−1(y) lies in one linearly connected component ofU ∩ f−1(y) for eachy ∈ f (V ).The mappingf is 1-soft if all its fibersf−1(y) are linearly connected and it is 1-soft ateach pointx ∈X.

Throughout, all spaces are assumed to be compact and all mappings continuous andsurjective.

Let A be a compact space. To each pair of surjective mappingss1 :B1 → A ands2 :B2→A of compact spacesB1 andB2 ontoA, we assign the diagramD(s1, s2):

B(s1,s2)

p2

p1B2

s2

B1s1

A

Here,B(s1,s2) = B1 ×(s1,s2) B2 = {(x, y) ∈ B1 × B2: s1(x) = s2(y)} is the fan product ofB1 andB2, andp1 andp2 are the natural projections.

Definition 2. Let F be a normal functor. We call a pair(b1, b2) ∈ F(B1)× F(B2) a softpair of the diagramF(D(s1, s2)) if there exists exactly one pointb ∈ F(B(s1,s2)) such thatF(p1)(b)= b1 andF(p2)(b)= b2.

M.V. Smurov / Topology and its Applications 107 (2000) 183–190 185

Let X be a compact space. We say that a pointa ∈ F(X) is F -soft if, for arbitrarycompact spacesB1, B2, and A, a mappings :A → X, surjectionss1 :B1 → A ands2 :B2→ A, and a pointa′ ∈ F(s)−1(a), each pair(b1, b2) ∈ F(B1) × F(B2) such thatF(s1)(b1)= F(s2)(b2)= a′ is a soft pair of the diagramF(D(s1, s2)).

Proposition 1. LetF be a functor of degreen. To find out whether or not a pointa ∈ F(X)is F -soft, it suffices only to verify that the requirements of Definition2 are fulfilled forsurjectionss :A→ suppa, s1 :B1→ A, ands2 :B2→ A of no more thann-point spacesA, B1, andB2.

Proof. Suppose given a mappings :A→X and surjectionss1 :B1→A ands2 :B2→A.Considera′ ∈ F(s)−1(a) and a pair(b1, b2) ∈ F(B1) × F(B2) such thatF(s1)(b1) =F(s2)(b2) = a′. Let s′, s′1, and s′2 denote the restrictions of the mappingss, s1, ands2to suppa′, suppb1, and suppb2, respectively. The pair(b1, b2) is soft in the diagramF(D(s1, s2)) if and only if it is soft in the diagramF(D(s′1, s′2)). 2Proposition 2. If F is a functor of degreen and anya ∈ F(X) with a one-point support isF -soft, thenF coincides with thenth power functorIdn.

Proof. If the spaceA in the diagramD(s1, s2) is a singleton, thenB(s1,s2) = B1 × B2.The condition that points with one-point supports are soft implies thatF(B(s1,s2)) =F(B1×B2)= F(B1)×F(B2) for such diagrams. Thus, for any compact spacesX andY ,F(X× Y )= F(X)× F(Y ); sinceF has degreen, F = Idn [3, Theorem 3.3]. 2Proposition 3. If F is of degreen and the support of a pointa ∈ F(X) comprisesn points,thena is anF -soft point.

Proof. An arbitrary surjections :A→ suppa is one-to-one, and so are any surjectionssi :Bi → A (i = 1,2) of no more thann-point spacesBi ontoA; therefore, all mappingsin the diagramsD(s1, s2) andF(D(s1, s2)) are one-to-one, which implies thata is F -soft. 2

Let X be a compact space,F a functor of degreen, anda a point inF(X). By Ba ,we denote the set of all familiesu= {Ux : x ∈ suppa} of neighborhoodsUx of x ∈ suppaopen inX and having pairwise disjoint closuresUx . For u ∈ Ba , r(u,a) :⋃{Ux : Ux ∈u} → suppa is the mapping defined byr(u,a)(Ux) = x, and 〈u,a〉F is the set{b ∈F(X): suppb⊂⋃u andF(r(u,a))(b)= a}.

Proposition 4. For a finitary functorF of finite degree and an arbitrary compact spaceX,the family{〈u,a〉F : u ∈ Ba} forms a base of the topology ofF(X) at the pointa ∈ F(X).

Proof. Let supp :F(X)→ expX map each pointa ∈ F(X) into its support suppa treatedas an element of the hyperspace expX. Recall that the hyperspace expX of a compactspaceX is the set of closed subsets ofX with the Vietoris topology. The base of the Vietoris


topology consists of the sets〈u〉 = {F ∈ expX: F ⊂⋃u, F ∩U 6= ∅ for all U ∈ u}, whereu are finite families of open subsets ofX.

If u ∈ Ba , then the set〈u,a〉F is open inF(X). Indeed, this set is the intersection ofsupp−1〈u〉 andF(r(u,a))−1(a); the second term is open inF(

⋃{U : U ∈ u}), and the firstone is contained inF(

⋃{U : U ∈ u}) and open inF(X), because the mapping supp iscontinuous for finitary functors [3].

Note that, if two neighborhoods belong to the system of neighborhoods ofa describedabove, then their intersection also belongs to this system. Therefore, to prove that thissystem forms a base of the topology ofF(X) at a, it remains to show that, for anypoint a′ ∈ F(X) such thata 6= a′, a has a neighborhood〈u,a〉F whose closure does notcontaina′. If the support ofa′ contains a pointx /∈ suppa, then it is sufficient to selectuso thatx /∈⋃{U : U ∈ u}. If suppa′ ⊂ suppa, then the closure of any neighborhood of theform 〈u,a〉F does not containa′. 2

2. A criterion for local 1-softness ofF(f )

Let F be a finitary functor of degreen, f :X → Y a 1-soft mapping of compactmetric spaces, andY a compact space without isolated points. Suppose thatb1 ∈ F(X)anda = F(f )(b1). Let B1 andA denote the supports ofb1 anda, respectively, ands1the restrictionf �B1 of f to B1. Note that the normality of the functorF implies thats1(B1)=A.

Proposition 5. Under the assumptions made above, the following conditions are equiva-lent:

(1) the mappingF(f ) is 1-soft atb1;(2) for any surjections2 :B2→ A of a no more thann-point spaceB2 ontoA and an

arbitrary b2 ∈ F(B2) such thatF(s2)(b2)= a, the pair(b1, b2) is a soft pair of thediagramF(D(s1, s2)).

Proof. In proving Proposition 5, we use the following notation. Let〈u,b1〉F be the baseneighborhood of a pointb1 in F(X) determined by a systemu= {Ux : x ∈ B1, x ∈ Ux} asdescribed just before Proposition 4. We say that a mappings2 :B2→A defined on a closedsubsetB2 of Y is consistent withu if y ∈ f (Ux) for any pair(x, y) ∈ B1×(s1,s2) B2. For amappings2 :B2→A consistent withu, p(u,s2) :C→ B1×(s1,s2) B2 denotes the mappingdefined on

C =⋃{

f−1(y)∩Ux : (x, y) ∈ B1×(s1,s2) B2}

by p(u,s2)(x) = (r(u,b1)(x), f (x)). Note that the consistency ofs2 and u implies thesurjectivity of the mappingp(u,s2). By q(u,s2), we denote an arbitrary right inverse top(u,s2).The discreteness of the spaceB1×(s1,s2) B2 implies thatq(u,s2) is continuous.

Let us prove the implication(1)⇒ (2). Take a base neighborhood (in the sense ofProposition 4)U of b1 in F(X) determined by a systemu = {Ux : x ∈ B1}. Since themappingF(f ) is 1-soft, there exists a neighborhoodV = 〈v, b1〉F of b1 (determined by


a systemv = {Vx : x ∈ B1, Vx ⊂ Ux}) such that any set of the formV ∩ F(f )−1(b) liesin one linearly connected component of the setU ∩ F(f )−1(b). Because the mappingsf andF(f ) are open, we can find a neighborhoodW = 〈w,a〉F of a in Y such thatW ⊂ F(f )(V) andWy ⊂ f (Vx) whenevery = f (x).

Let s2 :B2→ A be an arbitrary surjection of a no more thann-point spaceB2 ontoAand b2 ∈ F(B2) a point such thatF(s2)(b2) = a. Since the compact spaceY does nothave isolated points, we can embedB2 into

⋃w and assume thats2 = r(w,b2) �B2. The

mappings2 is consistent withu, and the setU ∩ F(f )−1(b2) is nonempty. Suppose thatc ∈ U ∩F(f )−1(b2); then suppc lies inC (the domain of the mappingp = p(u,s2)). By thedefinition ofp, F(pi)◦F(p)(c)= bi for i = 1,2; here,pi :B(s1,s2)→Bi are the mappingsof the diagramD(s1, s2). Thus, the set{b ∈ F(B(s1,s2)): F(p1)(b)= b1, F(p2)(b)= b2}is nonempty. Letd1 andd2 be two elements of this set. Denote the mappingq(u,s2) by qand putci = F(q)(di). We can assume thatq(B(s1,s2))⊂

⋃v. The equalityp ◦ q = id �B

implies thatci ∈ V ∩F(f )−1(b2) for i = 1,2. By the choice of the neighborhoodV , thereexists a mappinge : [0;1] → U such thate(0) = c1, e(1) = c2, andF(f )(e(t)) = b2 forall t ∈ [0;1]. The sets suppe(t) lie in C; therefore, the mappingF(p) ◦ e is defined for allt ∈ [0;1]. Since the spaceF(B(s1,s2)) is finite, the mappingF(p) ◦ e is constant. Note thatF(p)(ci)= di ; this givesd1= d2.

Thus, the set{b ∈ F(B(s1,s2)): F(p1)(b) = b1, F(p2)(b) = b2} is a singleton, and(b1, b2) is a soft pair of the diagramF(D(s1, s2)).

Let us prove that(2)⇒ (1). Take a base neighborhood (in the sense of Proposition 4)U of b1 in F(X) determined by a systemu= {Ux : x ∈ B1}. For eachx ∈ B1, by virtue ofthe 1-softness off , we can take an open neighborhoodVx of x in X such thatVx ⊂ Uxand the setVx ∩ f−1(y) lies in one linearly connected component of the intersectionUx ∩ f−1(y) for all y ∈ f (Vx). Denote the system{Vx : x ∈ B1} by v. Reducing, whennecessary, the neighborhoodsVx , we can assume that, forx1, x2 ∈ B1, f (x1) 6= f (x2)

implies f (Vx1) ∩ f (Vx2) = ∅, andf (x1) = f (x2) implies f (Vx1) = f (Vx2). For y ∈ A,let Wy denote the setf (Vx), wherex is an arbitrary element ofB1 ∩ f−1(y); this set isopen inY . The familyw = {Wy : y ∈ A} generates the neighborhoodW = 〈w,a〉F of ain F(Y ).

First, we prove thatW ⊂ F(f )(U). Take an arbitraryb2 ∈W and suppose thatB2 =suppb2. Put s2 = r(w,a) �B2, p = p(u,s2), q = q(u,s2), andB = B1 ×(s1,s2) B2. Sinces2is consistent withv, we can assume thatq(B) ⊂⋃v. As s1(b1) = s2(b2) = a and thepair (b1, b2) is soft, there existsb0 ∈ F(B) such thatF(pi)(b0) = bi for i = 1,2. Putb= F(q)(b0). It is easy to verify thatb ∈ V ∩ F(f )−1(b2).

Now, letb′ be an arbitrary element of the intersectionV ∩F(f )−1(b2). The mappingpsends suppb′ into B; in addition, if b′0 = F(p)(b′), thenF(pi)(b′0) = bi for i = 1,2.Since the pair(b1, b2) is soft in the diagramF(D(s1, s2)), we haveb′0= b0. Therefore, ford = (q ◦p)� suppb′, we haveF(d)(b′)= b andd(suppb′)= suppb. Sincep ◦ q = id �B,we also havep◦d = p � suppb′. This means that, for anyx ∈ suppb′, the pointsx andd(x)lie in the same element of the systemv and in the same layer of the mappingf .According to the properties of the systemv, for eachx ∈ b′, there exists a mappinghx : [0;1] →⋃

u such thathx(0) = x, hx(1) = d(x), and the range of the mappinghx


lies in one element of the systemu and in one fiber of the mappingf . Consider themappingH : suppb′ × [0;1] → ⋃

u defined byH(x, t) = hx(t). It is continuous, andH(x,0)= x, H(x,1)= d(x), (r(u,b1) ◦ H)(x, t)= r(u,b1)(x), and(f ◦H)(x, t) = f (x).Let us define a continuous family of mappingset : suppb′ → suppb′×[0;1] by the formulaet (x)= (x, t). The mappinge : I→ F(X) specified ase(t)= F(H ◦ et )(b′) is continuous,ande([0;1]) ∈ U ∩ F(f )−1(b2). The application of the equalitiese(0)= b′ ande(1)= bcompletes the proof of the 1-softness ofF(f ) atb1. 2Theorem 1. Suppose thatf :X→ Y is a1-soft mapping of compact metric spaces,Y hasno isolated points,F is a finitary functor of degreen, a ∈ F(Y ), and the fibersf−1(y) areinfinite for all y ∈ suppa. Then the following conditions are equivalent:

(1) there exists a closed neighborhoodU of a such that the mappingF(f )�F(f )−1(U)

is 1-soft;(2) the pointa is F -soft.

Proof. Suppose that condition (1) is fulfilled. By Proposition 4, there exists a neigh-borhoodW ⊂ U of a of the formW = 〈w,b〉F . Since the fibersf−1(y) of the pointsy ∈ suppa are infinite and the mappingf is 1-soft, we can assume that the neighbor-hoodsWy are selected so that the fibersf−1(y) are infinite for ally ∈⋃w. Consider anarbitrary surjections :A→ suppa of no more thann-point spaceA onto suppa. Sincethe setsW ∈ w are infinite, we can embedA into

⋃w in such a way thats = r(w,a) �A

and, therefore,F(s)−1(a) ∈W . Suppose thatsi :Bi → A with i = 1,2 are mappings ofno more thann-point spacesBi ontoA and(b1, b2) ∈ F(B1)× F(B2) is a pair such thatF(s1)(b1)= F(s2)(b2)= a′, whereF(s)(a′)= a. Let us embed the spaceB1 into f−1(A)

so thats1 = f �B1; Proposition 5 then implies that(b1, b2) is a soft pair of the diagramF(D(s1, s2)). By Proposition 1, the pointa is F -soft. This completes the proof of theimplication(1)⇒ (2).

Now, suppose thata ∈ F(Y ) satisfies condition (2). Take a base (in the sense ofProposition 4) neighborhoodW = 〈w,a〉F . Considerb1 ∈ F(f )−1(W) and denoteF(f )(b1) by a′. Let s be the restriction of the mappingr(w,a) to A = suppa′. ThenF(s)(a′)= a. Since the pointa is F -soft, Proposition 5 implies that the mappingF(f ) is1-soft atb1. Thus, the mappingF(f ) is 1-soft at any point of the preimageF(f )−1(W).This proves the implication(2)⇒ (1). 2

3. Functors and nonmetrizableAE(1)-compacta

Definition 3. ThedefectFd of a finitary functorF of finite degree is a subfunctor ofFdefined as follows: for a compact spaceX, the spaceFd(X) consists of all non-F -softpointsa ∈ F(X).

The correspondenceX→ Fd(X) defined above is a covariant functor in the category ofcompact spaces; this is immediately seen from the following easy lemma.


Lemma. For a compact spaceX and a finitary functorF of finite degree, the set{a ∈ F(X): a is F -soft} is open inF(X). If the image of the pointa ∈ F(X) under themappingF(f ), wheref :X→ Y is a mapping of compact spaces, isF -soft, then thepointa is alsoF -soft.

The next assertion follows from Propositions 2 and 3 and the observation that asubfunctor of a normal finitary functor of finite degree is a normal finitary functor of finitedegree.

Proposition 6. If a finitary functorF of a finite degreen is not thenth power functorIdn,then the functorFd is a finitary functor of degree lower thann.

Theorem 2. Let F andG be nonpower finitary functors of finite degree. Suppose thatfi :Xi→ Yi (i = 1,2) are 1-soft mappings of compact metric spaces,Yi have no isolatedpoints, and all fibers of the mappingsfi are infinite. If the mappingsF(f1) andG(f2)

are homeomorphic via homeomorphismshX :F(X1)→G(X2) andhY :F(Y1)→G(Y2),thenhY (Fd(Y1))=Gd(Y2).

Proof. Theorem 1 describesF -soft andG-soft points of the spacesF(Y1) andG(Y2) interms of the topology of the mappingsF(f1) andG(f2). Therefore, the mappinghY takesF -soft points of the spaceF(Y1) into F -soft points of the spaceG(Y2). 2

Let F be a finitary functor of finite degree. Using Proposition 6, we define a sequenceF1, . . . ,Fk(F ) (k(F ) is a positive integer) of its subfunctors such thatF1 = F , Fi+1 =(Fi)d , andFk(F ) is a power functor. We say thatk(F ) is thelengthof the functorF .

Theorem 3. Suppose thatX1 andX2 are nonmetrizable AE(1)-compacta homogeneouswith respect to character,F andG are finitary functors of finite degree, andh :F(X1)→G(X2) is a homeomorphism. Then the functorsF andG have the same lengthk, and, forall i = 1, . . . , k, there exist homeomorphismshi :Fi(X1)→Gi(X2), whereFi andGi arethe functors from the decompositions ofF andG.

Proof. It is sufficient to prove the existence of a homeomorphismhd :Fd(X1)→Gd(X2).The existence ofh shows that the spacesX andY are compact spaces of the same weightτ .The theorem on decomposability of nonmetrizableAE(1)-compacta into inverse systemsof compact metric spaces with 1-soft bonding mappings [2] and the Shchepin spectraltheorem [3] allow us to assert that there exist inverse systemsSi (i = 1,2) of compactmetric spacesXiα with 1-soft bonding mappingspiβα :Xiβ → Xiα and the same directed

setA and an isomorphismH = {hα :F(X1α)→G(X2

α), α ∈A} of these systems such thatXi = lim←−Si andh= lim←−hα . The latter condition means thatG(p2

βα) ◦ hβ = hα ◦ F(p1βα)

for β > α andG(p2α)◦h= hα ◦F(p1

α) (here,piα :Xi→Xiα are the limit projections). Takea1 ∈ F(X1) and puta2= h(a1). There existsα ∈ A such that the restrictionspiα � suppaiare one-to-one; therefore, the pointa1 is F -soft (the pointa2 is G-soft) if and only if the


point b1 = F(p1α)(a1) is F -soft (the pointb2=G(p2

α)(a2) is G-soft). Since the compactspacesXi are homogeneous with respect to character and the mappings in the systemsSi

are monotone, we can findβ ∈ A such that the fibers of allpiαβ are infinite. Applying

Theorem 2 to the mappingspiβα and the homeomorphismshα andhβ , we see that the

conditionsb1 ∈ Fd(X1α) andb2 ∈Gd(X2

α) are equivalent; therefore, so are the conditionsa1 ∈ Fd(X1) anda2 ∈Gd(X2). 2Corollary 1. If finitary functorsF and G of finite degree have different lengths andX1 andX2 are nonmetrizable AE(1)-compacta homogeneous with respect to character,then the spacesF(X1) andG(X2) are nonhomeomorphic.

Corollary 2. Suppose that nonmetrizable AE(1)-compactaX1 andX2 are homogeneouswith respect to character, satisfy the equalitiesX1= (X1)

ω andX2= (X2)ω (this is so if,

e.g., they are powers of Peano continua), and are nonhomeomorphic. Then, for any finitaryfunctorF of finite degree, the spacesF(X1) andF(X2) are also nonhomeomorphic.

Proof. Assume thatF(X1) andF(X2) are homeomorphic. According to Theorem 3, thespacesFk(X1) andFk(X2), wherek is the length ofF , are homeomorphic. By definition,Fk is a power functor, which completes the proof.2

References

[1] V.V. Fedorchuk, Covariant functors in a category of compacta, absolute retracts andQ-manifolds,Soviet Math. Surveys 36 (3) (1981) 177–195.

[2] V.V. Fedorchuk, On open mappings, Uspekhi Mat. Nauk 37 (4) (1982) 187–188 (in Russian).[3] E.V. Shchepin, Functors and uncountable powers of compacta, Soviet Math. Surveys 36 (3)

(1981) 1–71.[4] M.V. Smurov, On topological inhomogeneity of spaces of expKτ type, Soviet Math. Dokl. 22

(3) (1980) 716–721.[5] M.V. Smurov, Homeomorphisms of spaces of probability measures of uncountable degrees of

compacta, Soviet Math. Surveys 38 (3) (1983) 153–154.

functors and 1-soft mappings

Documents