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COVARIANT FUNCTORS IN CATEGORIES OF TOPOLOGICAL SPACES
M. M. Zarichnyi and V. V. Fedorchuk
UDC 515.12
This survey is devoted to the properties of certain concrete covariant functors --normal and almost normal functors --in the category of compacta, as well as the algebraic theory of covariant functors, and the connections between the theory of functors with absolute extensors and manifolds.
Category methods began to penetrate general topology shortly after the basic notions of category theory appeared. It turned out that many classical topological constructions, among the foremost of which are Tychonoff products and hyperspaces, have a functorial nature. At the present time it is possible to distinguish two approaches to application of the theory of categories to topology. The first consists in systematic investigation of axiomatically defined categories more general than the Top category of topological spaces and continuous transformations (the so-called topological categories). For a survey on this topic, the reader is referred to Herrlich's large article [232]. The second approach is associated with the study of the properties of functors acting in the Top category or its various subcategories. The most important general problem in this area is that of determining how the properties of spaces and transformations are modified upon application of covariant functors. The results obtained pertain not only to classical functors, but to a rather general class of functors, among which the most important is that of normal functors in the sense of E. V. Shchepin, operating in categories of compacta. At the present time, the normal functors constitute a class of functors for which it has been possible to construct a theory sufficiently complete to find a variety of algebraic and geometric applications. In this paper, which is based on articles that have been abstracted in the abstract journal "Matematika" (Mathematics) between 1953 and 1989, we present not only results on concrete functors (products, hyperspaces, superextensions), but also results on normal and almost normal functors. Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 28, pp. 47-95, 1990. 0090-4104/91/5302-0147512.50 9
Plenum Publishing Corporation
147
w PRODUCTS AND SYMMETRIC PRODUCTS 1. The construction of a Tychonoff product requires two essentially different types of functors: the power functor ( , ) a and the product functor (--) x A on the topological space A. Of course, these functors may be treated as endofunctors in each subcategory of the category Top that is closed under multiplication (in this case A is taken from the same subcategory). The most important of these subcategories are the categories Top/, i = 0,1,2,3 of the Ti-spaces, and Tych and Comp. Our choice of cited work on products was somewhat arbitrary. It was simply not possible within the framework of the present survey to cite all the relevant literature. There is literature on the effects of taking products (especially for raising to a power) on various general topological properties: normalness [56], [157], [289], [336], [340], [341], pseudocompactness [155], countable compactness [56], [201], final compactness [56], [157], [289], [ 336], [340], [341], pseudocompacmess [155], countable compactness [56], [201], final compactness [56], [289], [340], [341], paracompactness [289], [340], sequentialness [373], and others as noted in [22], [46], [69], [125], [128], [138], [156], [192], [225], [241], and [374]. A series of papers has been devoted to the effects of multiplication on cardinal-valued invariants [55], [56], [375], the dimensions functions dim, ind, Ind, and others [3], [64], [108], [113], [335], [339]. Van Mill [297] constructed a rigid metrizable space whose square is topologically homogeneous. A continuum of such examples was provided by Dijkstra [173]. V. V. Uspenskii [383] showed that an arbitrary space X can be converted into a topologically homogeneous space by taking the product with an appropriate space Y. Concerning the products of topologically homogeneous spaces, see [156]. Nonhomeomorphic countable spaces with homeomorphic squares were constructed by Marjanovic and Cucemilovic [285]. V. Trnkova [380] showed that if a countable metrizable space X is homeomorphic to Xn for n > 2, then X = X 2. Van Mill [297] presented an example of a Peano continuum X that is homeomorphic to X 2 by not homeomorphic to X '~ Khadzhiivanov [109], [110] and Todorov [376] studied properties like connectivity in finite and countable products of continua. In [110] it was shown that the countable product of non-single-point continua cannot be represented as the union of a countable sequence of closed subsets whose pairwise intersections are weakly infinite dimensional. Further information on products may be found in [142], [194], [183], [329], and [381]. 2. Product functors are closely associated with G-symmetric power functors, which we now describe. Let G be a subgroup of the group Sn of permutations of the set n = {0, 1..... n -- 1}. The group G operates on the n-th power of a space X by permuting its coordinates; the space of orbits of this operation is denoted by SGnXand is called the G-symmetric power of the space X. We denote the orbit of each point (x0. . . . . x.,,-t) ~X"- by Ix0. . . . . x,~-i]a. By defining, for any mappingf: X -, II, the mapping SPo'~T, SPa'~f Ix0. . . . . . xn-i]a ~ [r (xo) . . . . . T(x,~-t) ] o , we obtain the G-symmetric power functor SP~n defined in the category Top. Certain general properties of the spaces SGnXwere established in [396]. In [332] it was shown that shape properties of compacta such as (uniform) mobility, pointwise 1-mobility, and the property of being an absolute shape (neighborhood) retract are preserved by the G-symmetric power functors. A number of papers have been devoted to the homological properties of the spaces SPGnX [140], [176]-[178], [291], [322], [369], [370]. Thus, Dold showed that the homologies H,t(SPa."X, A), i ~ k , of a G-symmetric product depend only on the homologies H~ (X, A:), i~__2, if any k-element subfamily has a nonempty intersection. Assume N~,X={,.r162exp2X t,~t is k-chained and for all B exp X if B~A6.s~, then B05~}. For the mappingsf : X -, Ywe set Nhf(..~) ={Be exp YIB~f(A), Aosr The functors Nk, k >_ 2, were defined by A. V. Ivanov [44]. The retraction of N2X onto 2X was described in [301]; this retraction is not a natural mapping of X. Finally, E. V. Moiseev [61] defined the functor G of closed inclusion hyperspaces. By definition, GX ={~r exp2Xt for eachB ~ expX, if B~A~.sr then Bo~z~},Gf(,git)~-{B6 exp YIB~f(A), A~5~} (/'is a mapping f r o m X i n t o l 0. 151
The funetors 2 and N k, k > 2 are subfunctors of G. E. V. Moiseev [61] also introduced the functor G r o f closed growth hyperspaees, Gr X={ar exp~X[, if Ae.~ and B expX, B D A and any connected component o r b intersects A, then B~r Gr f(~r -----{Beexp Y[ B ~ f ( A ) and any connected component of B intersects A} (f: X -* Y is a morphism in Comp). A. V. Ivanov [38] studied the subfunctors An and *An of the superextension functor A. The space ~lnX consists of those ,r for which there exists a set C c X of degree _ ~ot, then exp X is not a dyadic compaetum (I.. B. Shapiro [114}). 162
2) for a nonmetrizable compactum X, the space exp X is neither an absolute retract nor even a continuous image of a Tychonoff cube (L. B. Shapiro [115]). 3) PX E A R if an only i f X ~ AE(O) and w(X) _< co1. if w(X) > oJt, then PX is not a diadic compactum (L. B. Shapiro [1161); 4) For a finite normal functor the following conditions are equivalent: i) F I~'.-~I~ ii) FD'~176 iii) F--- ( - - ) " for some n < co (E. V. Shchepin [1211). The following two propositions combine results of E. V. Shchepin and theorems on the form of multiplicative functors [33] (see w 5) for a normal functor F the following conditions are equivalent:
i) m'~'~AE{O); ii) F-~(--)", l ~ , ~ c o ; 6) if a normal functor F preserves the class of absolute retracts, then F ~ ( - - ) % l ~ z ~ o . A. G. Savchenko [84], [87] showed that if F is a functor of finite degree and the space FX is the continuous image of D ~ (respectively/t), the (connected) compactum X has the same property. M. Bell [132] proved the space exp D O'' is not the continuous image of a supercompact space. A. Ch. Chigogidze [152], assuming the continuum hypothesis, showed that the spaces D '~ and exp D ~ are not Baire isomorphic. In [323] and [20] there are results on the existence of spectral decompositions of a special form for nonmetrizable absolute exp-valued retracts and strong Dugundji spaces (i.e., absolute P~,-valued retracts), respectively. It was proved in [20] that if Kt is a strong Dugundji space for a metrizable compactum K and 9 > o~1, then K ~ AR. Superextension functors applied to nonmetrizable compacta behave considerably differently than normal functors. The following results were obtained by A. V. Ivanov: 7) kX~AE(0), if and only i f X is openly generated [39]; 8) 2X E A R if and only i f X is an openly generated continuum [40]. It follows from 7) and 8), together with characterizations of Cantor and Tychonoff cubes obtained by E. V. Shchepin [121], that XD'-~D', kl'~l" (A. V. Ivanov [41]) Assertions 7) and 8) were extended by A. V. Ivanov [44] and E. V. Moiseev [60], [61] to other superextension type functors. M. V. Smurov [90], [91] considered the problem of topological homogeneity of spaces of the form FX. THEOREM ([91]). Let h be an automorphism of the space PK ~, where K is a metrizable continuum, and r > co1. Then
h (P,K') -----P,K'. An analogous result was proved in [90] for the functor exp. E. V. Shehepin's method of characteristics [121] is essential to this proof. For normal functors of finite degree it has proved possible to obtain a complete solution to the problem of preservation of topological homogeneity. THEOREM 0M. M. Zarichnyi, see [35]). A normal functor of finite degree n preserves the class of topologically homogeneous compacta if and only if Fm ( - - ) " . Recently, results have been obtained on the geometry of mappings of nonmetrizable compacta. We will say that a mapping f : X --, Y is dyadic if all of its fibers are dyadic compacta. A mapping f : X -~ Y is said to be a Tychonoff bundle (/~ is a fiber) if it is homeomorphic to the projection pr, : Y>(I'--~-Y. THEOREM (V. V. Fedorchuk [101]). If, for an open mappingf: X --- Yof a compactum X onto a dyadic compactum Y, the closure of the set {y6 Y I IV-' (y) I >t2} is not metrizable, then the mapping Pf is not dyadic. THEOREM (V. V. Fedorchuk [101]). Let f : X ~ Y be a mapping of a nonmetrizable compactum X onto a dyadic compactum Y. The mapping P f is a Tychonoff bundle if and only if all of the three following conditions are satisfied: a) Y is a metrizable compactum; b) Xr and X is a compactum of weight co1 that is homogeneous with respect to character; c) f is open. As the following assertions show, the functors Jl, N, and G can considerably improve the properties of mappings of nonmetrizable compacta. 163
THEOREM ( ~ V. Ivanov, see [94]). For a mappingf : X * Yof openly generated continua that are homogeneous with respect to character, the following conditions are equivalent: i) f is open and has no points of homogeneity; ii) 2f is an/v-fiber, lr = w(X); iii) Nf is a D~-fiber. THEOREM (A. V. Ivanov, see [94]). For a mapping f : X .--, Y of zero-dimensional openly generated compacta that are homogeneous with respect to character, the following conditions are equivalent: i) f is open and has no points of multiplicity one; ii) 2fis a D~-fiber, 9 = w(X); iii) Nf is a D~-fiber. THEOREM (E. V. Moiseev [61]). Let f : X --, Y be a mapping of openly generated continua. The following conditions are equivalent: i) f is open; ii) Gf is soft. LITERATURE CITED .
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HILBERT MANIFOLDS WITH CORNERS OF FINITE CODIMENSION AND T H E T H E O R Y O F O P T I M A L C O N T R O L
UDC 517.974+517.977+515.164.174
S. A. Vakhrameev
This survey presents a version of Palais--Smale theory for Hilbert manifolds that is convenient for investigation of optimal control problems associated with smooth control systems of constant rank. Results obtained with A. A. Agrachev concerning the finite-dimensional case -- the analog of Morse theory for (finite-dimensional) manifolds with corners are given. Simple applications of the theory are discussed. A necessary condition for global controllability of systems of constant rank is obtained, as well as a dual result on the multiplicity of solutions for the corresponding optimization problems.
This article contains a complete exposition of results that were partially announced in papers by the author [12], and by the author and A. A. Agrachev [6], [49]; it is concerned primarily with one possible variant of Palais--Smale theory [62], [65], as it applies to the study of optimal control problems. The theory must be modified because of the special requirements of optimal control problems: In contrast to problems in the classical calculus of variations, optimal control problems have constraints in the form of inequalities; e.g., constraints on control action, on phase coordinates, on the right ends of admissible trajectories, etc. The inequalities upset the smooth structure of the problem, so it is necessary to consider spaces with singularities included in the very statement of the optimal control problem. In w of this article we consider classical Morse and Palais--Smale theory in connection with optimal control problems. Here we consider a class of controlled systems (systems of constant rank) that, to some extent, limits the applicability of theory to optimal control. Later, however, we show that this class is also a natural candidate for the modifications proposed in w Systems of constant rank were introduced by A.A. Agrachev and the author in [4], [5] (see also [11], [13]) in connection with research on conditions under which extremal control is two-position control; these systems constitute a rather broad class of controlled systems that are of interest for application in problems concerning the control systems that appear in classical mechanics. Moreover, in w we show that for a functional T
l I~ x, u)dt, 0
defined on the trajectories of a linear-control system of constant rank m
X.--'--f(x)+Zttg,(x),
xfiM,
l=!
Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 28, pp. 96-171, 1990. 176
0090-4104/91/5302-0176512.50
01991 Plenum Publishing Corporation