--in the category of compacta, as well as the algebraic theory of covariant .... Goodykoontz [207] constructed a nonlocally connected continuum X that is a retract ...
299. 300. 301. 302. 303. 304. 305. 306. 307. 08,
309. 310. 311. 312. 313. 314.
H. Subramanian, "Integer-valued continuous functions. II," Can. Math. Bull., 14, No. 3, 235-237 (1971). I. Suciu, Function Algebras, Acad. RSR, Bucharest (1975). R. P. Sullivan, "Continuous nilpotents on topological spaces," J. Austral. Math. Sot., 43, No. 1, 127-136 (1987). E. S. Thomas, Jr., "Spaces determined by their homeomorphism groups," Trans. Am. Math. Soc., 126, No. 2, 244-250 (1%7). M. C. Thornton, "Semirings of functions determine finite TO topologies," Proc. Am. Math. Sot., 34, No. 1, 307-310 (1972). M. C. Thornton, "Characterizing topologies by functions," Proc. Am. Math. Soc., 36, No. 2, 613-614 (1972). W. J. Thron, "Lattice-equivalence of topological spaces," Duke Math. J., 29, No. 4, 671-679 (1%2). W. J. Thron, "Proximity structures and grills," Math. Ann., 206, 35-62 (1%3). R. L. Vanght, "The continuous real-valued functions on a completely regular space," Bull. Am. Math. Soc., 55, No. 7, 716 (1949). A. R. Vobach, "On subgroups of the homeomorphism group of the Cantor set," Fund. Math., 60, No. 1, 47-52 (1967). A. D. Wallace, "Some characterization of interior transformations," Am. J. Math., 41, 757-763 (1939). H. Wallman, "Lattices and topological spaces," Ann. Math. 39, No. 1, 112-126 (1938). J. C. Warndof, "Topoloies uniquely determined by their continuous self-maps," Fund. Math., 66, No. 1, 25-43 (1969). M. T. Wechsler, "Homeomorphism groups of certain topological spaces," Ann. Math., 66, No. 3, 360-373 (1955). J. V. Whittaker, "On isomorphic groups and homeomorphic spaces," Ann. Math., 78, No. 1, 74-91 (1%3). C. L. Wiginton and S. Shrader, "Spaces determined by a group of functions," Bull. Am. Math. Soc., 74, No. 6, 1110-1112 (1970).
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 .
2. 3. 4. 5. .
7.
8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
164
~
B. V. Atamanyuk, "Functors of iterated superextension," Vestn. Mosk. Gos. Univ., Mat. Mekh., No. 3, 58 (1989). T. O. Banakh and M. M. Zarichnyi, "On compactifications of iteratated hyperspace functors," Izv. Vyssh. Uchebn. Zaved., Mat., No. 10, 3-6 (1987). V. N. Basmanov, "On inductive dimensionality of product spaces," Vestn. Mosk. Gos. Univ., Mat. Mekh., No. 1, 17-20 (1981). V. N. Basmanov, "Some functors with finite support and dimensionality," Vesm. Mosk. Gos. Univ., Mat. Mekh., No. 6, 48-50 (1981). V. N. Basmanov, "Covariant functors, retracts, and dimensionality," Dokl. AN SSSR, 271, No. 5, 1033-1036 (1983). V. N. Basmanov, Functors Transforming Connected ANR-Bicompacta into Singly Connected Spaces, VINITI report, No. 2781-83. V. N. Basmanov, "Functors transforming connected ANR-bicompacta into singly connected spaces," Vestn. Mosk. Gos. Univ., Mat. Mekh, No. 6, 40-42 (1984). V. N. Basmanov, "Covariant functors of finite degree and connectivity," Dold. AN SSSR, 279, No. 6, 1289-1293 (1984). S. A. Bogatiyi and V. V. Fedorchuk, "Theory of retracts and infinite-dimensional manifolds," Itogi Nauki i Tekhniki VINITI. Algebra. Topologiya, Geometriya, No. 24, 195-270 (1986). A. N. Vybornov, Hyperspaces of Complete Metric Separable Zero-Dimensional Spaces, VINITI report, No. 7912-84. A. N. Vybornov, "Spaces of compact subsets of locally bicompact zero-dimensional separable metric spaces," in: Maps and Functors [in Russian], Moscow State Univ. (1984), pp. 17-23. A. N. Vybornov, "Extension of homeomorphisms and exponents of zero-dimensional locally compact metrizable spaces," in: Cardinal Invariants and Mappings of Topological Spaces [in Russian], Izhevsk (1984), pp. 80-81. A. N. Vybornov, Classification of the Exponents of Locally Compact Metric Zero-Dimensional Spaces, VINITI report, No. 7911-84. A. N. Vybornov, "Spaces of compact subsets of zero-dimensional Polish spaces with everywhere dense sets of isolated points," Usp. Mat. Nauk, 39, No. 4, 153-154 (1984). A. N. Vybornov, "Exponents of zero-dimensional Polish spaces," Dokl. AN SSSR, 284, No. 5, 1053-1057 (1985). V. I. Golov, "Hilbert spaces as hyperspaces of ordered arcs," in: Functional Analysis and Its Applications in Mechanics and Probability Theory [in Russian], Moscow (1984), pp. 13-18. V. I. Golov, "Ordered arecs of Peano continua. A layer variant," Vestn. Mosk. Gos. Univ., Mat. Mekh, No. 3, 1318(1984). A. N. Dranishnikov, "Mixers. The converse of a theorem of van Mill and van de Vel," Mat. Zametki. 37, No. 4, 587-593 (1984). A. N. Dranishnikov, "Covariant functors and extensors in n dimensions," Lisp. Mat. Nauk, 40, No. 6, 133-134 (1985). A. N. Dranishnikov, "Absolute F-valued retracts and spaces of functions in the topology of pointwise convergence," Sib. Mat. Zh., 27, No. 3, 74-86 (1986).
21. 22. 23. 24. 25. 26. 27.
29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54.
A. N. Dranishnikov, "On Q-stratification without disjunctive cuts," Funk. Anal. Prilozhen., 22, No. 2, 79-80 (1988). L. G. Zambakhidze, "Interrelations between local boundary properties of topological spaces," Soobshch. AN GruzSSR, 94, No. 3, 557-560 (1979). M. M. Zarichnyi, "The functors ;ln and absolute retracts, ~ Vestn. Mosk. Gos. Univ., Mat. Mekh., No. 4, 15-19 (1982). M. M. Zarichnyi, Subfunctors of Superextension Functors, VINITI report, No. 316-82. M. M. Zarichnyi, nHypersymmetric powers of supercompacta," Vestn. Mosk. Gos. Univ., Mat. Mekh., No. 1, 18-21 (1983). M. M. Zarichnyi, "Preservation of :ANR(~)-spaces and infinite-dimensional manifolds by some covariant functors," Dokl. AN SSR, 271, No. 3, 524-528 (1983). M. M. Zarichnyi, "The fundamental group of superextensions ;lnX," in Mappings and Functors [in Russian], Moscow Slate Univ. (1984), pp. 24-31. M. M. Zarichnyi, nOn a result of J. van Mill andA. ShriveL" Vestn. Mosk. Gos. Univ., Mat. Mekh., No. 2, 6-8 (1985). M. M. Zarichnyi, Visn. L'viv. Univ. Ser. Mekh. Mat., No. 24, 65-69 (1985). M. M. ~qfichnyi, Visn. L'viv. Univ. Ser. Mekh. Mat. No. 25, 52-56 (1986). M. M. Zarichnyi, "Iterated superextensions," in: General Topology. Mappings of Topological Spaces [in Russian], Moscow State Univ. (1986), pp. 45-59. M. M. Zarichnyi, "Monadic functors of finite degree,~ in: Problems of Geometry and Topology [in Russian], Petrozavodsk, (1986), pp. 24-30. M. M. Zarichnyi, "A Multiplicative normal functor is a power functor," Mat. Zametld, 41, No. 1, 93-100 (1987). M. M. Zarichnyi, "The monad of superextensions and its algebras," Ukr. Mat. Zh., 39, No. 3, 303-309 (1987). M. M. Zarichnyi, Visn. L'viv. Univ. Ser. Mekh. Mat. No. 30, 30-32 (1988). M. M. 7_,arichnyiand L.P. Plakhta, "Normal functors of finite degree on the category PL," Dokl. AN Ukr. SSR, No. 9, 59 (I988). A. V. Ivanov, "Superextensions of metrizable continua and the generalized Cantor discontinuum," Dokl. AN SSSR, 254, No. 2, 279-281 (1980). A. V. Ivanov, *On the superextension 2n," Usp. Mat. Nauk, 36, No. 3, 213-214 (1981). A. V. Ivanov, "Superextensions of openly generated bicompacta," Dokl. AN SSSR, 259, No. 2, 275-278 (1981). A. V. Ivanov, "A solution to van Mill's problem on characterizations of compacta whose superextensions are absolute retracts," Dokl. AN SSSR, 262, No. 3, 526-528 (1982). A~ V. Ivanov, "The supperextension/r of a Tychonoff cube is homeomorphic to/r,, Mat. Zametki, 33, No. 5, 763-772 (1983). A. V. Ivanov, "Saperextension functors and smooth maps," Mat. Zametki, 36, No. 1, 1984. A. V. Ivanov, "A theorem on almost fixed points for mappings of spaces of maximal K-chained systems," in" Problems of Geometry and Topology [in Russian], Petrozavodsk (1986), pp. 31-40. A. V. Ivanov, "Spaces of complete chained systems," Sib. Mat. Zh., 27, No. 6, 95-110 (1986). N. L. Koval', "Peano-valued absolute retracts," Vestn. Mosk. Gos. Univ., Mat. Mekh., No. 1, 3-5 (1987). A. P. Kombarov, "Hereditary paracompactness o f X 2 and metrizability of X," Vestn. Mosk. Gos. Univ., Mat. Mekh., No. 2, 79-81 (1988). V. O. Kuznetsov, "A space of coles subsets," Dokl. AN SSSR, 178, No. 6, 1248-1251 (1968). R. S. Lintschuk, "The dimensionality of hyperspaces, * Ukr. Mat. Zh., 30, No. 3, 378-381 (1978)1. V. O. Kuznetsov, "Dyadic spaces in certain hyperspaces, ~ Ukr. Mat. Zh., 30, No. 2, 232-234 (1978). V. O. Kuznetsov, "Hyperextensions of set-valued maps and spectral representation of certain hyperspaces," Ukr. Mat. Zh., 33, No. 2, 248-252 (1981). V. O. Kuznetsov, "The properties of some hyperspaces of infinite-dimensional type," Ukr. Mat. Zh., 34, No. 6, 774-775 (1982). V. O. Kuznetsov and V. M. Safonov, "Exponential topology," Ukr. Mat. Zh., 35, No. 2, 164-168 (1983). V. O. Kuznetsov and V. I. Sirik, "On the dimensionality of spaces of subsets," in: Function Theory and Topology [in Russian], Kiev (1983), pp. 68-76. V. O. Kuznetsov and V. I. Sirik, "On the dimensionality of hyperspaces," in: Function Theory and Topology [in Russian], Kiev (1983), pp. 76-83.
165
55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86.
166
V. I. Malykhin, "Tightness and the Suslin number in exp X in in product spaces," Dokl. AN SSSR, 203, No. 5, 1001-1003 (1972). V. I. Malykhin, "Properties lost by topological groups when they are squared," Sib. Mat. Zh., 28, No. 4, 154-161 (1987). M. M. Mar'yanovich, "Higher-order hyperspaces," Dokl. AN SSSR, 278, No. 1, 34-37 (1984). R. I~. Mirzakhanyan, "Infinite iteration of the inclusion-hyperspace functor," Vestn. Mosk. Gos. Univ., Ser. 1, No. 6, 1417 (1988). R. t~. Mirzakhanyan, "A functor for continuation of the inclusion hyperspace," Vesm. Mosk. Gos. Univ., Ser. 1, No. 2, 75-77 (1989). E. V. Moiseev, "Spaces of closed growth and inclusion hyperspaces," Vestn. Mosk. Gos. Univ., Mat. Mekh, No. 3, 54-57 (1988). E. V. Moiseev, Functors of Closed Growth and Inclusion Hyperspaces, VINITI report, No. 3515-B89. M. Mrshchevich, "Properties of the space 2X of the topological R0-space," Usp. Mat. Nauk, 34, No. 6, 166-170 (1979). G. M. Hepomnyashchii, "Spectral decomposition of manifolds of absolute retracts," Usp. Mat. Nauk, 36, No. 3, 221-222 (1981). B. A. Pasynkov, "The dimensionality of topological products and limits of inverse sequences," Dokl. AN SSSR, 254, No. 6, 1332-1336 (1980). A. Pelczynski, Linear Continuation, Linear Extension, and Their Application to Linear Topological Classification of Spaces of Continuous Functions [Russian translation], Mir, Moscow (1970). L. P. Plakhta, "Preservation of homologies prime homotopic type of normal functors," AN USSR Preprint, Inst. Mat., No. 54, 1-28 (1988). L. P. Plakhta, "Preservation of manifolds by functors of finite degree," Int. Prikl. Probl. Mekh. Mat. AN SSSR Preprint, No. 5, 1-16 (1988). V. I. Ponomarev, "Absoluteness of spaces and their exponents," Vestn. Mosk. Gos. Univ., Mat. Mekh., No. 2, 59-62 (1987). V. I. Ponomarev and V. V. Tkachuk, "A countable character X in fix compared with a countable character of the diagonal o f X • X," Vestn. Mosk. Gos. Univ., Mat. Mekh., No. 5, 16-19 (1988). V. V. Popov, "The topology of spaces of closed subsets," Vestn. Mosk. Gos. Univ., Mat. Mekh., No. 1, 65-70 (1975). V. V. Popov, "A space of closed subsets," Dokl. AN SSSR, 229, No. 5, 1051-1054 (1976). V. V. Popov, "Exponents, X-products, and k-spaces," in Topological Spaces and Their Mappings [in Russian], Riga (1979), pp. 107-118. V. V. Popov, "Metrizability and spaces of closed subsets," Usp. Mat. Nauk, 35, No. 3, 209-213 (1980). V. V. Popov, "Normality of the exponent in Ochan topologies," Mat. Zametki, 32, No. 3, 375-384 (1982). V. V. Popov, "Exponents of Y~-products of topological spaces," in: Topology and Set Theory [in Russian], Izhevsk (1982), pp. 20-25. V. V. Popov, "Normality of exponents in topogies of Ochanov type. II," Lit. Mat. Sb., 23, No. 4, 32-39 (1984). V. V. Popov, "The structure of exponents of discrete spaces," Mat. Zametki, 35, No. 5, 757-767 (1984). V. V. Pepov, "Exponents in Ochanov Topologies and k-spaces," in: Cardinal Invariants and Mappings of Topological Spaces [in Russian], Izhevsk (1984), pp. 19-22. V. V. Popov, "Spaces of bicompact subsets and k-spaces," in" General Topology. Function Spaces and Dimensionality [in Russian], Moscow State Univ. (1985), pp. 121-130. G. O. Praninskas, "Countable compactness and pseudocompactness of exponents in Ochanov type topopologies," Lit. Mat. Sb, 27, No. 2, 344-347 (1985). I. E. Prokhorov, "Hypermappings of continua," Mat. Zametki, 42, No. 4, 572-580 (1987). A. G. Savchenko, "The properties of the mapping exPnC],"Vestn. Mosk. Gos. Univ., Mat. Mekh., No. 1, 19-25 (1985). A. G. Savchenko, "The functor expnc, absolute retracts, and Hilbert spaces," Mat. Zametki, 38, No. 6, 875-887 (1985). A. G. Savchenko, "Functors of finite degree: criteria for isomorphism of spower functors and bicompacta that are continuous images of/v and D~," VINITI report No. 7888. A. G. Savchenko, "Some covariant functors and dimensionality," in: General Topology. Mappings of Topological Spaces [in Russian]; Moscow State Univ. (1986), pp. 104-121. A. G. Savchenko, "Covariant functors of finite degree, dimension and compacta of full dimensionality," Usp. Mat. Nauk, 41, No. 4, 221-222 (1986).
87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. i18. 119. 120. 121. 122. 123.
A.G. Savchenko, "On bicompacta of the form F(X) that are continuous images of/v and Dr," Vestn. Mosk. Gos. Univ., Mat. Mekh, No. 3, 3-5 (1988). A.G. Savchenko, ~Criteria for isomorphism of functors of finite degree and power functors, n Vestn. MosZ Gos. Univ., Ser. 1, No. 3, 18-21 (1989), S.O. Sirota, "Sp~tral representation of spaces of closed subsets of bicompacta," Dold. AN SSSR, 181, No. 5, 1169-1172 (1968). M.V. Smurov, "On topological inhomogeneity of spaces of type exp KL Dokl. AN SSSR, 255, No. 3, 526-531 (1989). M.V. Smirov, "Homeomorphisms of spaces of probability measure for uncountable powers of compacta," Usp. Mat. Nauk, 38, No. 3, 187-188 (1983). U. Tashmetov, "On connectivity of hyperspaces," Dokl. Akad. Nauk SSSR, 215, No. 2, 286-288 (1974). U. Tashmetov, "Connectivity and local connectivity of certain hyperspaces, ~ Sib. Mat. Zh., 15, No. 5, 1115-1130 (1974). Tirasporskii, Symposium on General Topology and Its Applications [in Russian], Shtinitsa, Kishinev (1985). V.V. Fedorchuk, "Probability measures and absolute retracts," Dokl. Akad. Nauk SSSR, 255, No. 6, 1329-1333 (1980). V.V. Fedorchuk, "Covariant functors in catgories of compacta, absolute retracts, and Q-manifolds," Usp. Mat. Nauk, 36, No. 3, 177-190 (1981). V.V. Fedorchuk, nExponents of Peano continua -- the layer variant," Dokl. Akad. Nauk SSSR, 262, No. 1, 41-44 (1982). V.V. Fedorchuk, "On open mappings," Usp. Mat. Nauk, 37, No. 4, 187-188 (1982). V.V. Fedorchuk,"Monotonic open mappings," Publ. Inst. Math. 34, 61-64 (1983). V.V. Fedorchuk, "Some geometric properties of covariant functors," Usp. Mat. Nauk, 39, No. 5, 169-208 (1984). V.V. Fedorchuk, "Trivial stratification of spaces of probability measure," Mat. Sb. 129, No. 4, 473-493 (1986). V.V. Fedorchuk, "On characterization of n-soft mappings," Dokl. Akad. Nauk SSSR, 28g, No. 2, 313-316 (1986). V.V. Fedorchuk, "Soft maps, set-valued retractions, and functors," Usp. Mat. Nauk, ,11, No. 6, 12159 (1986). V.V. Fedorchuk, "A factorization lemma for open mappings of compacta," Mat. Zametki, 42, No. 1, 101-114 (1987). V.V. Fedorchuk, "Set-valued retractions and characterization of n-soft mappings," Tr. Mosk. Mat. O-va, 51, 169-207 (1988). V.V. Fedorchuk, "Stratification of spaces of probability measure and the geometry of infinite iterations of certain monadic functors," Dokl. Akad. Nauk SSSR, 301, No. 1, 41-45 (1988). V.V. Fedorchuk and V. V. Filippov, General Topology. Basic Constructions [in Russian], Moscow State Univ. (1988). V.V. Filippov, "The dimension of products of topological spaces," Fund. Math., 106, No. 3, 181-212 (1980). N.G. Khadzhiivanov, "On products of continua," Dokl. Bulg. Akad. Nauk., 31, No. 10, 1241-1244 (1978). N.G. Khadzhiivanov, "On the degree of connectivity of products of continua," Serdika. B"ulg. Mat. Spicanie, 5, No. 4, 380-383 (1979). T. Chapman, Lectures on Q-Manifolds [Russian translation], Mir, Moscow (1981). A. Ch. Chigogidze, "Continuation of normal functors," Vestn. Mosk. Gos. Univ., Mat. Mekh., No. 6, 23-26 (1984). A~ Ch. Chigogidze and B. A~ Pasynkov, "The dimensionality of products of completely regular topological spaces," Soobshch. Akad Nauk Gruz. SSR, 90, No. 3, 553-556 (1978). L.B. Shapiro, "The space of closed subsets of D a2 is not a dyadic bicompactum," Dokl. Akad. Nauk SSSR, 228, No. 6, 1302-1305 (1976). L.B. Shapiro, "Spaces of closed subsets of bicompacta," Dokl. Akad. Nauk SSSR, 231, No. 2, 295-298 (1976). L. B2 Shapiro, "Spaces of probability measure," Usp. Mat. Nauk, 34, No. 2, 2219-220 (1979). L.B. Shapiro, "Spaces of absolute dyadic bicompacta," Dokl. Akad. Nauk SSSR, 293, No. 5, 1077-1081 (1987). L.B. Shapiro, "The category characteristic of a hyperspace," Usp. Mat. Nauk, 43, No. 4, 227-228 (1988). E.V. Shchepin, "Real-valued functions and canonical sets in Tychonoff products and topological groups, n Usp. Mat. Nauk, 31, No. 6, 17-27 (1976). E . V . Shchepin, "Topological limits of spaces of uncountable inverse spectra," Usp. Mat. Nauk, 31, No. 5, 191-226 (1976). E.V. Shchepin, "Functors and uncountable powers of compacta," Lisp. Mat. Nauk, 36, No. 3, 3-62 (1981). E: V. Shchepin, "The method of inverse spectra in the topology of bicompacta," Mat. Zametki, 31, No. 2, 299-315 (1981). F. Albrecht, "Asupra spatiului multimilor inchise al unui spatiu compact," Commun. Acad; R, P. Romane, 3, Nos. 1-2, 13-17 (1953). 167
124. 125. 126. 127. 128. 129. 130. 131. 132. 133. 134. 135. 136. 137. 138. 139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150. 151. 152. 153. 154. 155. 156. 157. 158. 159. 160. 161. 162.
168
R.D. Anderson and R. H. Bing, "A complete elementary proof that Hilbert space is homeomorphic to the countable infinite product of lines," Bull. Am. Math. Soc., 74, No. 5, 771-792 (1968). Y. Aoki, "Orthocompactness of inverse limits and products," Tsukuba J. Math., 4, No. 2, 241-255 (1980). G. Artico and R. Moresco, "Notes on topologies and uniformities of hyperspaces," Rend. Sere. Univ. Padova, 63, 51-60 (1980). P.C. Baayen, "Maximal linked systems in topology," in: Gen. Topoi. and Relat. Modern Anal. and Algebra IV. Pro. 4th Prague Topoi. Syrup., Part B, Prague (1976), pp. 28-36. R.W. Bagley and D. D. Weddington, "Product of k'-spaces," Proc. Am. Math. Soc., 22, No. 2, 292-394 (1969). C. Bandt, "On the metric structure of hyperspaces with Hausdorff metric," Math. Nachr., 129, 175-183 (1986). M. Barr and C. Wells, Toposes, Triples, and Theories, Springer, New York (1985). M. Bell, "Hyperspaces of finite subsets," Math. Centre Tracts. Amsterdam, No. 115, 15-28 (1979). M. Bell, "Supercompactness of compactifications and hyperspaces," Trans. Am. Math. Soc. (as in Russian original). M. Bell, J. Ginsburg, and S. Todor~vi~, "Countable spread of exp Y and 2Y," Topoi. Appl., 14, No. 1, 1-12 (1982). C.R. Borges, "Normality of hyperspaces," Math. Jpn., 25, No. 4, 507-510 (1980). C. Borges, "Retraction properties of hyperspaces," Math. Jpn., 30, No. 4, 551-557 (1985). C. Borges, "Hyperconnectivity of hyperspaces," Math. Jpn., 30, No. 5, 757-761 (1985). C. Borges, "LEC's, local mixers, topological groups and special products," 3. Austral. Math. Soc., 44, No. 2, 252-258 (1988). J. Borsik and J. Dobog, "On a product of metric spaces,' Math. Slovaca (CSSR), 31, No. 2, 193-205 (1981). R. Bott, "On the third symmetric potency of $1, " Fundamen. Math., 39, No. 3, 264-268 (1952). R. Bott, "On symmetric products and Steenrod squares," Ann. Math., Ser. 2, 57, No. 3, 579-590 (1953). L. Boxer, "Hyperspaces where convergence to a calm limit implies eventual shape equivalence," Fund. Math., 115, No. 3, 213-222 (1983). M.A. Brown, "A note on cartesian produts," Am. J. Math., 91, No. 1, 32-36 (1969). R. Cauty, "Produits sym6triques de r6tractes absolus de voisinage," C. R. Acad. Sci. Paris, 276, No. 5, 359-361 (1973). J.J. Charatonik, "Contractibility and continuous selections," Fund. Math. 108, No. 2, 109-118 (1980). W.J. Charatonik, "Hyperspaces and the property of Kelley," Bull. Acad. Pol. Sci. S6r. Sci. Math., 30, Nos. 9-10, 457-459 (1982). W.J. Charatonik, "On the property of Kelley in hyperspaces," Lect. Notes Math., 1060, 7-10 (1984). W.J. Charatonik, "Homogeneity is not a Whitney property," Proc. Am. Math. Soc., 92, No. 2, 311-312 (1984). W.J. Charatonik, "A continuum X which has no confluent Whitney map for 2x," Proc. Am. Math. Soc., 92, No. 2, 313314 (1984). W.J. Charatonik, "Pointwise smooth dendroids have contractible hyperspaces," Bull. Pol. Acad. Sci. Math., 33, Nos. 7-8, 409-412 (1985). W.J. Charatonik, "A metric on hyperspaces defined by Whitney maps," Proc. Am. Math. Soc., 94, No. 3, 535-538 (1985). Z. (;erin and A. P. Sostak, "Fundamental and approximative uniformity on the hyperspace," Glas. Math., 16, No. 2, 339359 (1981). A. Chigogidze, "On Baire isomorphisms of non-metrizable compacta," Comment. Math. Univ. Carol., 26, No. 4, 811-820 (1985). A. Chigogidze and V. Valov, "Set-valued maps and AE(0)-spaces," Dokl. Bolg. Akad. Nauk, 41, No. 4, 13-14 (1988). M. Coban, "Note sur topologie exponentielle," Fund. Math., 71, No. 1, 27-41 (1971). W.W. Comfort, "A nonpseudocompact product space whose finite subproducts are pseudocompact," Math. Ann., 170, No. 1, 41-44 (1967). W. W. Comfort and J. van Mill, "On the product of homogeneous spaces," Topoi. Appl., 21, No. 3, 297-308 (1985). A. Csazar, "Normality in products," in: Topology, 4th Cooloq., Budapest, 1978, Vol. 1, Budapest (1980). D.W. Curtis, "The hyperspace of subcontinua of Peano continuum," Lect. Notes Moth., 378, 108-118 (1974). D.W. Curtis, "Growth hyperspaces of Peano continua," Trans. Am. Math. Soe., 238, 271-283 (1978). D . W . Curtis, *Hyperspaces homeomorphic to Hilbert space," Proc. Am. Math. Soc., 75, No. 1, 126-130 (1979). D.W. Curtis, 'Hyperspaces of Peano continua," Math. Centre Tracts. Amsterdam, No. 115, 51-65 (1979). D.W. Curtis, "Hyperspaces of noncompact metric spaces, ~ Compos. Math., 40, No. 2, 139-152 (1980).
163. 164. 165. 166. 167. 168. 169. 170. 171. 172. 173. 174. 175. 176. 177. 178. 179. 180. 181. 182. 183. 184. 185. 186. 187. 188. 189. 190. 191. 192. 193. 194. 195. 196. 197.
D . W . Curtis, "Application of a selection theorem to hyperspaces contractibility," Can. J. Math., 37, No. 4, 747-759 (1985). D.W. Curtis M. Michael, "Boundary sets for growth hyperspaces," Top. Appl., 25, No. 3, 269-283 (1987). D.W. Curtis and Nguyen To Nhu, "Hyperspaces of finite subsets which are homeomorphic to l~0-dimensional linear metric spaces," Top. Appl., 19, No. 3, 251-260 (1985). D.W. Curtis, J. Quinn, and R. M. Schori, "The hyperspace of compact convex subsets of a polyhedral 2-cell," Houston J. Math., 3, No. 1, 7-15 (1977). D.W. Curtis and R. M. Schori, "2X and C(X) are homeomorphic to the Hilbert cube," Bull. Am. Math. Soc., 80, No. 5, 927-931 (1974). D.W. Curtis and R. M. Schori, "Hyperspaces which characterize simple homotopy type," Gen. Topoi. Appl., 6, No. 2, 153-165 (1976). D.W. Curtis and R. M. Schori, "Hyperspaces of polyhedra are Hilbert cubes," Fund. Math., 99, No. 3, 189-197 (1978). D.W. Curtis and R. M. Schori, "Hyperspaces of Peano continua are Hilbert cubes," Fund. Math., 101, No. 1, 19-38 (1978). J.M. Day and K. Kuratowski, "On the nonexistence of a continuous selector for arcs lying in the plane," Indag. Math., 28, No. 2, 131-132 (1966). P. Diamont and P. Kloeden, "A note on compact sets in spaces of subsets," Bull. Austral. Math. Soc., 38, No. 3, 393-395 (1988). J.J. Dijkstra, "A rigid space whose square is the Hilbert space," Proc. Am. Math. Soc., 93, No. 1, 118-120 (1985). S.Z. Ditor and L. Q. Eifler, "Some open mapping theorems for measures," Trans. Am. Math. Soc., 164, 278-293 (1972). S.Z. Ditor and R. Haydon, "On absolute retracts, P(S), and complemented subspaces of C(Dw),"Stud. Math., 56, No. 3, 243-251 (1976). A. Dold, "Homology of symmetric products and other functors of complexes," Ann. Math., 68, No. 1, 54-80 (1958). A. Dold, "Decomposition theorem for S(n)-complexes," Ann. Math., 75, No. 1, 8-16 (1962). A. Dold and D. Puppe, "Homologie nicht-additiver Funktoren," Andwendungen. Ann. Inst. Fourier, 11, 301-312 (1961). A. Dold and R. Thom, "Quasifaserungen und unendliche symmetrische Produkten," Ann. Math. Ser 2, 67, No. 2, 239-281 (1958). C. Dorsett, "Local connectedness, connectedness im kleinen, and other properties of hyperspaces of R 0 spaces," Mat. Vestn., 3, No. 2, 113-132 (1979). C. Dorsett, "Connectedness in hyperspaces," J. Natur. Sci. Math., 22, No. 1, 61-70 (1982). C. Dorsett, "Local connectedness in hyperspaces," Rend. Circ. Mat. Palermo, 31, No. 1, 137-144 (1982). C. Dorsett, "Product spaces and semi-separation axioms," Period. Math. Hung., 13, No. 1, 39-45 (1982). C. Dorsett, "Connectivity properties in hyperspaces and product spaces," Fund. Math., 121, No. 3, 189-197 (1984). E . K . van Douwen and J. T. Goodykoontz, Jr., "Aposyndesis in hyperspaces and ~.ech--Stone remainders," In: Gen. Topoi. and Mod. Anal. Proc. Conf., Riverside, Calif. May 28-31, 1980, New York (1981), pp. 43-52. M.M. Dregevi~, "On inverse limit of functions spaces," Publ. Inst. Math., 15, 47-54 (1973). C. Eberhart, "Continua with locally connected Whitney continua," Houston J. Math., 4, No. 2, 165-173 (1978). C. Eberhart and S. B. Nadler, Jr., "The dimension of certain hyperspaces," Bull. Acad. Pol. Sci. S6r. Sci. Math. Astron. Phys., 19, No. 11, 1027-1034 (1971). C. Eberhart and S. B. Nadler, Jr., "Hyperspaces of cones and fans," Proc. Am. Math. Soc., 77, No. 2, 279-288 (1979). C. Eberhart and S. B. Nadler, Jr., "Irreducible Whitney levels," Houston J. Math., 6, No. 3, 355-363 (1980). C. Eberhart, S. B. Nadler, and W. O. Nowell, Jr., "Spaces of order arcs in hyperspaces," Fund. Math., 112, No. 2, 11-120 (1981). Eda Katsuya, "A note on the hereditary properties in the product space," Tsukuba J. Math., 4, No. 2, 157-159 (1980). S. Eilenberg and J. Moore, "Adjoint functors and triples," IlL J. Math., 9, No. 3, 381-398 (1965). F. van Engelen, "Countable products of zero-dimensional absolute Fo6-spaces," Proc. Koninkl. Nederl, Akad. Wet., 87, No. 4, 391-399 (1984). V.V. Fedorchuk, "On hypermaps which are trivial bundles," Lect. Notes. Math., 1060, 26-36 (1984). V.V. Fedorchuk, "Absolute retracts and some functors," in: Gen. Topoi. and Relat. Mod. Anal. and Algebra 5, Berlin, (1983), pp. 174-182. B. Fisher, "A reformulation of the Hausdorff metric," Rostock Math. Kolloq., No. 24, 71-76 (1983). 169
198. 199. 200. 201. 202. 203. 204. 205. 206. 207. 208. 209. 210. 211. 212. 213. 214. 215. 216. 217. 218. 219. 220. 221. 222. 223. 224. 225. 226. 227. 228. 229.
t-70:
K. Fort, Jr. and J. Segal, "Minimal representations of the hyperspace of a continuum," Duke Math. J., 32, No. 1, 129-137 (1965). S. Francaviglia, A. Lechnicki, and S. Levi, *Quasiuniformization of hyperspaces and convergence of nets of semicontinuous multifunctions," J. Math. Anal. Appl., 112, No. 2, 347-370 (1985). L.M. Friedler, R. F. Dickman, Jr., R. L. Krystock, "Hyperspaces of H-closed spaces," Can. J. Math., 32, No. 5, 1072-1079 (1980). Z. Frolik, "Sums of ultrafilters," Bull. Am. Math. Soc., 73, No. 1, 87-91 (1967). T. Ganea, *Contractibilitatea producelor simetrice," Stud. Cerc. Mat., 4, Nos. 1-2, 23-28 (1953). T. Ganea, "Symmetrische Potenzen topologischer R~iume," Math. Nachr., 11, Nos. 4-5, 305-316 (1954). T. Ganea, "Produse semitrice di spatii topologice," Commun. Acad. R. P. Romine, 4, Nos. 11-12, 561-563 (1954). R.J. Gazik, "Uniform convergence for a hyperspace," Proc. Am. Math. Soc., 42, No. 1, 302-306 (1974). J. Ginsburg, nA note on G~-Closure and the realcompactness of 2x," Gen. Top. Appl., 6, No. 3, 319-326 (1976). J.T. Goodykoontz, Jr., "Nonlocally connected continuum X such that C(X) is a retract of 2x," Proc. Am. Math. Soc., 91, No. 2, 312-322 (1984). J.T. Goodykoontz, Jr., "Aposyndetic properties of hyperspaces," Pac. J. Math., 47, No.l, 91-98 (1973). J.T. Goodykoontz, Jr., "Connectedness im kleinen and local connectedness in 2X and C(X)," Pac. J. Math., 53, No. 2, 387-398 (1974). J.T. Goodykoontz, Jr., "Some functions on hyperspaces of hereditarily unicoherent continua," Fund. Math., 95, No. 1, 110 (1977). J.T. Goodykoontz, Jr., "C(X) is not necessarily a retract of 2X" Proc. Am. Math. Soc., 67, No. 1, 177-178 (1977). J.T. Goodykoontz, Jr., "Hyperspaces of arc-smooth continua," Houston J. Math., 7, No. 1, 33-41 (1981). J.T. Goodykoontz, Jr., %rc-smootheness in hyperspaces," Topoi. Appl., 15, No. 2, 131-150 (1983). J.T. Goodykoontz, Jr., "Some retraction and deformation retraction on 2X and C(X)," Topoi. Appl., 51, No. 2, 121-133 (1985). J.T. Goodykoontz, Jr., "Geometric models of Whitney levels," Houston J. Math., 11, No. 1, 75-89 (1985). J.T. Goodykoontz, Jr. and S. B. Nadler, Jr., "Whitney levels in hyperspaces of certain Peano continua," Trans. Am. Math. Soc., 274, No. 2, 671-694 (1982). N.K. Gray, "Unstable points in the hyperspace of connected subsets," Pac. J. Math., 23, No. 3, 515-520 (1967). N. IC Gray, "On the conjecture 2X ~ 1%" Fund. Math., 66, No. 1, 45-52 (1969). J. Grispolakis, S. B. Nadler, Jr., and E. D. Tymchatyn, "Some properties of hyperspaces with applications to continua theory," Can. J. Math., 31, No. 1, 197-210 (1979). J. Grispolakis and E. D. Tymchatyn, "Embedding smooth dendroids in hyperspaces," Can. J. Math., 31, No. 1, 130-I38 (1979). J. Grispolakis and E. D. Tymchatyn, "Irreducible continua with degenerate end-tranches and arcwise accessibility in hyperspaces," Fund. Math., 110, No. 2, 11%130 (1980). J. Grispolakis and E. D. Tymchatyn, "On a characterization of W-sets and the dimension of hyperspaces," Proc. Am. Math. Soc., 100, No. 3, 557-563 (1987). J. de Groot, "On some problems of Borsuk concerning a hyperspace of compact sets," Proc. Koninkl. Akad. Wet., 29, No. 1, 95-103 (1956). J. de Groot, G. A. Gensen, and A. Verbeek, "Superextension," in: Proe. Int. Syrup. Topoi. and Its Appl. Herceg-Novi, 1968, Belgrad (1969), pp. 176-178. C.L. Hagopian, "Products of hereditarily indecomposable continua are Jl-connected," Fund. Math., 119, No. 3, 217-226 (1983). J.M. Harvey, "Categorical characterization of uniform hyperspaces," Math. Proc. Cambridge Phil. Sot., No. 2, 229-233 (1983). R. Haydon, *On problems of Pelczynski: Milutin spaces, Dugundji spaces, and AE(dim 0)," Stud. Math., 52, No. 1, 23-31 (1974). S.H. Hechler, "Exponents of some N-compact spaces," Isr. J. Math., 15, No. 4, 384-395 (1973). R.E. Heisey and J. E. West, "Orbit spaces of the hyperspace of a graph which are Hilbert cubes," Colloq. Math., 56, No. 1, 59-69 (1988).
230. 231. 232. 233. 234. 235. 236. 237. 238. 239. 240. 241. 242. 243. 244. 245. 246. 247. 248. 249. 250. 251. 252. 253. 254. 255. 256. 257. 258. 259. 260. 261. 262. 263. 264.
D.W. Henderson, *A simplicial complex whose product with any ANR is a simplicial complex, ~ Gen. Topoi. Appl., 3, No. 1, 81-83 (1973). D.W. Henderson, "On the hyperspace of subcontinua of an arc-like continuum," Proc. Am. Math. Soc., 27, No. 2, 416417 (1971). H. Herrlich, "Categorical topology," in: Gen. Topoi. and Relat. Mod. Anal. and Algebra. 5, Berlin (1983), pp. 279-383. A.G. Hitchcock, "Uniform hyperspaces and Hausdorff completions," J. London Math. Soc., 12, No. 4, 505-512 (1976). A. Hohti, "On supercomplete spaces. III," Proc. Am. Math. Soc., 97, No. 2, 339-342 (1986). M. Husek and J. Pelant, ~Extensions and restrictions in products of metric spaces," Top. Appl., 25, No. 3, 245-252 (1987). A.M. Illanes, "Multicoherence of symmetric products," An. Inst. Math. UNAM, 22, 11-24 (1985). A.M. Illanes, "Monotone and open Whitney maps," Proc. Am. Math. Soc., 98, No. 3, 516-518 (1986). A~ M. Illanes, "Irreducible Whitney levels with respect to finite and countable subsets," An. Inst. Mat. UNAM, 26, 59-64 (1986). A.M. Illanes, "A continuum X which is a retract of C(X) but not of 2X* Proc. Am. Math. Soc., 100, No. 1, 199-200 (1987). A.M. IUanes, *Cells and cubes in hyperspaces," Fund. Math., 130, No. 1, 57-65 (1988). T. Ishii, "A product formula for the Tychonoff functor," in: Topology. 4th Colloq. Budapest, 1978, Amsterdam (1980), pp. 659-661. J.W. Jaworowski, "Symmetric products of ANR's," Math. Ann., 192, No. 3, 173-176 (1971). J.W. Jaworowski, "Symmetric product of ANR's associated with a permutation group,' Bull. Acad. Pol. Sci., Ser. Math., 20, No. 8, 649-651 (1972). H. Kato, "Concerning hyperspaces of certain Peano continua and strong regularity of Whitney maps," Pac. J. Math., 119, No. 1, 159-167 (1985). H. Kato, "Shape properties of Whitney maps for hyperspaces, ~ Trans. Am. Math. Soc., 297, No. 2, 529-546 (1986). H. Kato, "The dimension of hyperspaces of certain 2-dimensional continua," Topoi. Appl., 28, No. 1, 83-87 (1988). H. Kato, ~On admissible Whitney maps," Colloq. Math., 56, No. 2, 299-309 (1988). H. Kato, "Shape equivalence of Whitney continua of curves,* Can. J. Math., 40, No. 1, 217-227 (1988). H. Kato, "Whitney continua of graphs admit all homotopy types of compact connected ANR's, * Fund. Math., 129, No. 3, 161-166 (1988). H. Kato, "Movability and strong Whitney-reversible properties, ~ Topoi. Appl., 31, No. 2, 125-132 (1989). J. Keesling, ~Normality and properties related to compactness in hyperspaces," Proc. Am. Math. Soc., 24, No. 4, 760-766 (1970). J. Kennedy, "Compactifying the space of homeomorphisms," Colloq. Math., 56, No. 1, 41-58 (1988). V. Klee, nSome topological properties of convex sets," Trans. Am. Math. Soc., 78, No. 1, 30-45 (1955). Y. Kodama, S. Spie~, and T. Watanabe, "On shape of hyperspaces," Fund. Math., I00, No. 1, 59-67 (1978). Shu-Chung Koo, "Recursive properties of transformation groups in hyperspaces," Math. Syst. Theory, 9, No. 1, 75-82 (1975). J. Krasinkiewics, "Certain properties of hyperspaces," Bull. Acad. Pol. Sci. Ser. Sci. Math. Astron. Phys., 21, No. 8, 705710 (1973). J. Krasinkiewics, "On Hyperspaces of snake-like and circle-like continua," Fund. Math., 83, No. 2, 155-164 (1973). J. Krasinkiewics, "On the hyperspaces of hereditarily indecomposable continua," Fund. Math., 84, No. 3, 175-186 (1974). J. Krasinkiewics, "On the hyperspaces of certain plane continua,* Bull. Acad. Pol. Sci. S6r. Sci. Math. Astron. Phys., 23, No. 9, 981-983 (1975). J. Krasinkiewics and S. B. Nadler, Jr., "Whitney properties, ~ Fund. Math., 98, No. 2, 165-180 (1978). N. Kroonenber, "Pseudointeriors of hyperspaces," Compos. Math., 32,. No. 2, 113-131 (1976). A.Y.W. Lau, ~A note on monotone maps and hyperspaces, ~ Bull. Acad. Pol. Sci. S6r. Sci. Math. Astron. Phys., 24, No. 2, 121-123 (1976). A.Y.W. Lau and C. H. Voas, "Connections of the hyperspaces of closed connected subsets," Rocz. Pol. Tow~ Mat., Ser. 1, 201 NO. 2, 393-396 (1978). J.D. Lawson, "Applications of topological algebra to hyperspace problems," in: Topology, New York--Basel (1976), pp. 201-206. 171
265. 266. 267. 268. 269. 270. 271. 272. 273. 274. 275. 276. 277. 278. 279. 280. 281. 282. 283. 284. 285. 286. 287. 288. 289. 290. 291. 292. 293. 294. 295. 296.
172
A. Lechnicki and S. Levi, "Wijsman convergence in the hyperspaces of metric space," Rend. Circ. Math. Palermo, 34, Suppl., No. 8, 297-299 (1986). A. l_~chnicki and S. Levi, "Wijsman convergence in the hyperspace of a metric space," Boll. Unione Mat. Ital., 1, No. 2, 439-451 (1987). S.D. Liao, "On the topology of cyclic products of spheres," Trans. Am. Math. Soc., 77, 520-551 (1954). I. Lon~ar, "Relative continuity of the functor fl," Czechosl. Math. J., 37, No. 4, 517-521 (1987). E. Lowen-Colebunders, "On the uniformization of the hyperspace of closed convergence," Math. Nachr., 105, 35-44 (1982). E. Lowen-Colebunders, "On the convergence of closed and compact sets," Pac. J. Math., 108, No. 1, 133-140 (1983). M. Lynch, "Whitney properties for 1-dimensional continua," Acad. Bull. Pol. Sci. S6r. Sci. Math., 35, Nos. 7-8, 473-478 (1987). M. Lynch, "Whitney levels in Cp(X) are AR's," Proc. Am. Math. Soc., 97, No. 4, 748-750 (1986). T. Mackowiak, "Contractable and nonselectable dendroids," Bull. Acad. Pol. Sci. S6r. Sci. Math., 33, Nos. 5-6, 321-324 (1985). T. Mackowiak, "Continuous selections for C(Y0," Bull. Acad. Pol. Sci. S6r. Sci. Math. Astron. Phys., 26, No. 6, 547-551 (1978). B.L. McAllister, "Hyperspaces and multi-functions. The first half century (1900-1950)," Nieuw. Arch. Wisk., 26, No. 2, 309-329 (1978). M.M. Marjanovi~, "Exponentially complete spaces I.," Glass. Math., 6, No. 1, 143-147 (1971). M.M. Marjanovi~, "Exponentially complete spaces," Publ. Inst. Math., No. 13, 77-79 (1972). M.M. Marjanovi6, "Exponentially complete space. III.," Publ. Inst. Math., No. 14, 97-109 (1972). M . M . Marjanovi6, "Spaces homeomorphic to their hyperspaces," In: Proc. Intern. Syrup. Topoi. and Appl., Budva, Belgrade (1973), pp. 167-169. M.M. Marjanovi~, "Exponentially complete spaces. IV," Publ. Inst. Math., No. 16, 101-109 (1973). M.M. Marjanovi6, "Some questions related to hyperspaces," In: Gen. Topoi. and Relat. Modern Anal. and Algebra IV. Proc. 4th Prague Topoi. Syrup. 1976., Part B, Prague (1977), pp. 276-278. M.M. Marjanovi6, "Spaces X ~' for zero-dimensional X," Bull. CI. Sci. Math. et Natur. Sci. Math./Acad. Serbe Sci. et Arts, 47, No. 16, 23-26 (1988). M.M. Marjanovi~ and S. T. Vrc~ica, "Another hyperspace representation of the Hilbert cube," Bull. Acad. Serbe. Sci. et Arts, 88, No. 14, 11-19 (1985). M.M. Marjanovi~, S. T. Vre~ica, and R. T. Zivaljevi~, "Some properties of hyperspaces of higher rank," Bull. Acad. Serbe Sci. et Arts, 84, No. 13, 103-117 (1984). M.M. Marjanovi6 and A. R. Cucemilovi6, "Two nonhomeomorphic countable spaces having homeomorphic squares," Comment. Math. Univ. Carol., 26, No. 3, 479-588 (1985). S. Masih, "Fixed points of symmetric product mappings of polyhedra and metric absolute neighborhood retracts," Fund. Math., 811, 149-156 (1973). C.N. Maxwell, "Fixed points of symmetric products mappings," Proc. Am. Math. Soc., 8, No. 4, 808-815 (1957). E. Michael, "On a map from a function space to a hyperspace," Math. Ann., 162, No. 1, 87-88 (1965). E. Michael, "Paracompactness and the Lindel0f property in finite and countable cartesian products," Compos. Math., 23, No. 2, 199-214 (1971). M. Michael, "Some hyperspaces homeomorphic to separable Hilbert space," In: Gen. Topoi. and Mod. Anal. Proc. Conf. Riverside, Calif., May 28-31, 1980, New York (1981), pp. 291-294. R. Milgram, "The homology of symmetric products," Trans. Am. Math. Soc., 138, 251-265 (1969). J. van Mill, Superextensions and Wallman Spaces, No. 75, Amsterdam (1977). J. van Mill, "Recent results on superextensions," In: Gen. Topoi. and Relat. Modern. Anal. and Algebra IV., Proc. 4th Prague Topoi. Syrup., 1976, Part. B., Prague (1977), pp. 279-283. J. van Mill, "A pseudo-interior of ;l. I," Compos. Math., 36, No. 1, 75-82 (1978). J. van Mill, "The superextension of the closed unit interval is homeomorphic to the Hilbert cube," Fund. Math., 1113,No. 3, 151-175 (1979). J. van Mill, "Superextensions of metrizable continua are Hilbert Cubes," Fund. Math., 107, No. 3, 201-224 (1980).
297. 298. 299. 300. 301. 302. 303. 304. 305. 306. 307. 308. 309. 310. 311. 312. 313. 314. 315. 316. 317. 318. 319. 320. 321. 322. 323. 324. 325. 326. 327. 328. 329. 330. 331.
J. van Mill, "A Peano continuum homeomorphic to its own square but not to its countable infinite product," Proc. Am. Math. Sot., 80, No. 4, 703-705 (1980). J. van Mill, "A rigid space X for which X x X is homogeneous, and application of infinite-dimensional topology," Proc. Am. Math. Soc., 83, No. 3, 597-600 (1981). J. van Mill and A. Schrijver, "Superextensions which are Hilbert cubes," Period. Math. Hung., 10, No. 1, 15-24 (1979). J. van Mill and M. van de Vel, "Path connectedness, contractibility, and LC-properties, of superextensions," Bull. Acad. Pol. Sci. S6r. Sci. Math. Astron. Phys., 26, No. 3, 261-269 (1978). J. van Mill, "On syperextensions and hyperspaces," Math. Centre Tracts. Amsterdam., No. 115, 169-180 (1979). J. van Mill, "Subbases, convex sets, and hyperspaces," Pac. J. Math., 92, No. 2, 389-402 (1981). D. Milovan~evi6, "Some relations between hyperspaces of nearly compact spaces," Zb. Rad. Fil. Fak. Nigu. Ser. Mat., 1, 55-59 (1987). A.K. Misra, "A note on arcs in hyperspaces," Acta Math. Hung., 45, Nos. 3-4, 285-288 (1985). T. Mizokami, "Cardinal functions on hyperspaces," Colloq. Math., 41, No. 2, 201-205 (1979). R. Molski, "On symmetric products," Fund. Math., 44, 165-170 (1957). P. Morales, "A note on the topology of C-convergence in hyperspaces," Proc. Am. Math. Soc., 52, 469-470 (1975). IC Morita, "Completion of hyperspaces of compact subsets and topological completion of open-closed maps," Gen. Topoi. Appl., 4, No. 3, 217-233 (1974). K. Morita, "On the dimension of the product of topological spaces," Tsukuba J. Math., 1, 1-6 (1977). H.N. Morton, "Symmetric products of the circle," Proc. Camb. Philos. Soc., 63, No. 2, 349-352 (1967). M. Mr~evi6, "On connectivity properties of hyperspaces of R0-spaces," Mat. Vestn., 3, No. 4, 451-458 (1979). S.B. Nadler, Jr., "Some problems concerning hyperspaces," Lect. Notes Math., 375, 190-197 (1974). S.B. Nadler, Jr., Some basic connectivity properties of Whitney map inverses in C(X)," in: Stud. Topoi., New York (1975), pp. 393-410. S.B. Nadler, Jr., "Continua whose cone and hyperspace are homeomorphic," Trans. Am. Math. Soc., 320, 321-345 (1977). S.B. Nadler, Jr., "A characterization of locally connected continua by hyperspace retractions," Proc. Am. Math. Soc., 67, No. 1, 167-176 (1977). S.B. Nadler, Jr., Hyperspaces of Sets. A Text With Research Questions, Marcel Dekker, New York--Basel (1978). S.B. Nadler, Jr., "Continua whose hyperspace is a product," Fund. Math., 108, No. 1, 49-66 (1980). S.B. Nadler, Jr., "Induced universal maps and some hyperspaces with the fixed point property," 100, No. 4, 749-754 (1987). S.B. Nadler, Jr. and J. Quinn, "Continua whose hyperspace and suspension are homeomorphic," Gen. Topoi. Appl., 8, No. 2, 119-126 (1978). S.B. Nadler, Jr., J. Quinn, and N. M. Stavrakas, "Hyperspaces of compact convex sets. II," Bull. Acad. Pol. Sci. S6r. Sci. Math. Astron. Phys., 25, No. 4, 381-385 (1977). S.B. Nadler, Jr., J. Quinn, and N. M. Stavrakas, "Hyperspaces of compact convex sets. II," Pac. J. Math., 83, No. 2, 441462 (1977). M. Nakaoka, "cohomology of the three-fold symmetric products of spheres," Pac. J. Math., 31, No. 10, 670-672 (1955). G. Nepomnjag~II, "A spectral characterization of absolute multivalued retracts," In: Gen. Topoi. and Relat. Mod. Anal. and Algebra. 5, Berlin (1983), pp. 507-510. Nguyen To Nhu, "Investigating the ANR-property of metric spaces," Fund. Math., 124, No. 3, 243-254 (1984). T. Nishiura and Choon-Jai Rhee, "Cut point of X and the hyperspace of subcontinua C(X)," Proc. Am. Math. Soc, 82, No. 1, 143-154 (1981). T. Nishiura and Choon-Jai Rhee, "Contractibility of the hyperspaces of subcontinua," Houston J. Math., 8, No. 1, 119127 (1982). Shin-ichi Nitta, "A note on superextensions of product spaces," Math. Jpn., 32, No. 1, 75-79 (1987). H. Otta, *Realcompactness of hyperspaces and extensions of open-closed maps," Topoi. Appl., 17, No. 3, 215-274 (1984). S. Oka, ~ychonoff functor and product spaces," Proc. Jpn. Acad., 54, No. 4, 97-100 (1978). A. Okuyama, "Note on hyperspaces consisting of compact sets," Math. Jpn., 24, No. 3, 301-305 (1979). A. Okuyama, *Some relationships between function spaces and hyperspaces, ~ In: Gen. Topoi. and Relat. Mod. Anal. and Algebra. 5, Berlin (1983), pp. 507-535. 173
332. 333. 334. 335. 336. 337. 338. 339. 340. 341. 342. 343. 344. 345. 346. 347. 348. 349. 350 351. 352. 353. 354. 355. 356. 357. 358. 359. 360. 361. 362. 363. 364. 365. 366. 367. 174
J. Oledski, *On symmetric products,* Fund. Math., 131, No. 3, 185-190 (1988). A. Pelczynski, *A remark on space 2X for zero-dimensional X,* Bull Acad. Pol. Sci. S6r. Sci. Math. Astron. Phys., 13, No. 2, 85-89 (1965). A. Petrus, *Contractibility of Whitney continua in C(X)," Gen. Topoi. Appl., 9, No. 3, 275-288 (1978). E. Pol, *On the dimension of the product of metrizable spaces," Bull. Acad. Pol. Sci. S6r. Sci. Math. Astron. Phys., 26, No. 6, 525-534 (1978). L. Polkowski, "On N. Noble's theorems concerning powers of spaces and mappings," Colloq. Math., 41, No. 2, 215-217 (1979). V. Popov, *On the subspaces of exp X," in: Topology. 4th Colloq. Budapest, 1978, 2, Amsterdam (1980), pp. 977-984. C.W. Proctor, "A characterization of absolutely C-smooth continua," Proc. Am. Math. Soc., 92, No. 2, 293-296 (1984). T.C. Przymusinski, "On the dimension of product spaces and an example of M. Wage," Proc. Am. Math. Soc., 76, No. 2, 315-321 (1979). T.C. Przymusinski, ~Normality and paracompactness in finite and countable Cartesian products,* Fund. Math., 105, No. 2, 87-104 (1980). T.C. Przymusinski, "Products of perfectly normal spaces," Fund. Math., 108, No. 2, 129-136 (1980). N. Rallis, "Periodic points of symmetric prduct mappings," Fund. Math., 117, No. 1, 61-66 (1983). N. Rallis, "A fixed point index theory for symmetric product mappings," Manuscr. Math., 44, No. 1, 279-308 (1983). C.J. Rhee, "On dimension of the hyperspace of pseduo-circle," Bull. Soc. R. Sci. Liege, 43, Nos. 1-2, 5-8 (1974). C.J. Rhee, "M-set and the contractibility of C(X)," Bull. Soc. R. Sci. Liege, 56, No. 7, 55-70 (1987). J.T. Rogers, Jr., "Dimension of hyperspaces," Bull. Acad. Pol. Sci. S6r. Sci. Math. Astron. Phys., 20, No. 2, 177-179 (1972). J, T. Rogers, Jr., "The cone-hyperspace property," Can. J. Math., 24, No. 2, 279-285 (1972). J.T. Rogers, Jr., "Whitney continua in the hyperspace C(X)," Pac. J. Math., 58, No. 2, 569-584 (1975). J.T. Rogers, Jr., "The Whitney subcontinua of C(X)," Bull. Acad. Pol. Sci. S6r. Sci. Math. Astron. Phys., 24, No. 2, 125127 (1976). J.T. Rogers, Jr., *Dimension and the Whitney subcontinua of C(X),* Gen. Topoi. Appl., 6, No. 1, 91-100 (1976). J.T. Rogers, Jr., "Applications of Vietoris--Begle theorem for multivalued maps to the cohomology of hyperspaces," Mich. Math. J., 22, No. 4, 315-319 (1976). H.J. Schmidt, "Hyperspaces topologies indued by extensions," In: Proc. Conf. Topol. and Meas. III, Vitte/Hiddensee, Oct. 19-25, 1980, Part 2, Greifswald (1982), pp. 251-257. H.J. Schmidt, "Hyperspaces of quotient and subspaces. I. Hausdorff topological spaces," Math. Nachr., 104, 271-280 (1981). H.J. Schmidt, "Hyperspaces of quotient and subspaces, II. Metrizable spaces,* Math. Nachr., 104, 281-288 (1981). H . J . Schmidt, *Topologies of the hyperspaces which are induced by extension spaces,* Math. Nachr., 118, 105-113 (1984). R.M. ISchori, *Hyperspaces and symmetric products of topological spaces,* Fund. Math., 63, No. 1, 77-88 (1968). R.M. Schori, "Inverse limits and near-homeomorphism techniques in hyperspace problems, * Lect. Notes Math., 378, 421-428 (1974). R. M. Schori and J. E. West, "21 is homeomorphic to the Hilbert cube," Bull. Am. Math. Soc., 78, No. 3, 402-406 (1972). R. M. Schori and J. E. West, "The hyperspace of the closed unit interval is a Hilbert cube," Trans. Am. Math. Soc, 213, 217-235 (1975). J. Segal, "Hyperspaces of the inverse limit space,* Proc. Am. Math. Soc., 10, No. 5, 706-709 (1959). J. Segal, "A fixed point theorem for the hyperspace of a snakelike continuum," Fund. Math., 50, No. 3, 237-248 (1962). M. Sekanina, *Hyperspaces and algebras," In: Proc. Conf. Topoi. and Meas. 2, Rostock-Warnemtide, Oct24-No. 2, 1977, Part I (Sect. A), Greifswald (1980), pp. 139-145. D.D. Sherling, *Concerning the cone-hyperspace proPerty," Can. J. Math., 35, No. 6, 1030-1048 (1983). D.H. Smith, *A note on complete hyperspaces,* Proc. Cambridge Phil. Soc., 52, No. 3, 602-604 (1956). D.H. Smith, "The Borel structure of a hyperspace," Proc. Cambridge Phil. Soc., 63, No. 4, 947-948 (1967), M. Smith, *Concerning the homeomorphisms of the p~seudo-carc X as a subspace of C(X x X),* Houston J. Math., 12, No. 3, 431-440 (1986). E. Spanier, *Infinite symmetric products, function spaces and duality,~ Ann. Math., 69, No. 1, 142-192 (1959).
368.
369. 370. 371. 372. 373. 374. 375. 376. 377. 378. 379. 380. 381. 382. 383. 384. 385. 386. 387. 388. 389. 390. 391. 392. 393. 394. 395. 396. 397. 398. 399. 400. 401. 402. 403. 404. 405.
M. Stanojevi~, "On hyperspace of compact subsets of k-spaces," Zb. Rad. Fil. Fak. Nigu. Ser. Mat. Univ. Nigu., 2, 55-60 (1988). S.K. Stein, "The homology of the two-fold symmetric product," Ann. Math., $9, No. 3, 570-583 (1954). R. Swan, "The homology of cyclic products," Trans. Am. Math. Soc., 95, 27-68 (1960). T. Swirszcz, "Monadic functors and convexity," Bull. Acad. Pol. Sci. S6r. Sci. Mat. Astron, Phys., 22, No. 1, 39-42 (1974). A. Szankowski, "Projective potencies and multiplicative extension operators," Fund. Math., 67, No. 1, 97-113 (1970). Y. Tanaka, "Products of sequential spaces," Proc. Am. Math. Soc., 54, 371-375 (1976). Y. Tanaka and Hao-Xuan Shou, "Products of closed images of CW-complexes and k-spaces," Proc. Am. Math. Soc., 92, No. 3, 465-469 (1984). S. Todorc~vie, "Remarks on cellularity in products," Comp. Math., 57, 357-372 (1986). V.T. Todorov, "Remarks on cellularity in products," Dokl. Bolg. AN SSSR, 39, No. 5, 5-8 (1986). H. Toruficzyk, "On CE-images of the Hilbert cube and characterization of Q-manifolds," Fund. Math., 106, No. 1, 39-40 (1980). H. Toruficzyk and J. West, "A Hilbert space limit for the iterated hyperspace functor," Proc. Am. Math. Soc., 89, No. 2, 329-335 (1983). W . R . R . Transue, "On the hyperspace of subcontinua of the pseudoarc," Proc. Am. Math. Soe., 18, No. 6, 1074-1075 (1967). V. Trnkova, "Homeomorphisms of powers of metric spaces," Comment. Math. Univ. Carol., 21, No. 1, 41-53 (1980). K. Tsuda, "Some examples concerning the dimension of product spaces," Mat. Jpn., 27, No. 2, 177-195 (1982). M. Turzfisld, "Remarks on superextensi0ns," Colloq. Math., 43, No. 2, 243-249 (1980). E . D . Tymchatyn, "Hyperspaces of hereditarily indecomposable plane continua," Proc. Am. Math. Soc., 56, 300-302 (1976). V. Uspensldi, "For any X, the product X • Y is homogeneous for some Y," Proc. Am. Math. Soc., 87, No. 1, 187-188 (1983). V.M. Valov, "Another characterization ofAE(0)-spaces," Pac. J. Math., 127, No. 1, 199-208 (1987). V.M. Valov, "Milutin mappings and AE(0)-spaces," Dokl. Bulg. Akad. Nauk, 40, No. 11, 9-12 (1987). V.M. Valov, "Yet another characterization ofAE(0)-spaces," Proc. 17th Prolet. Conf. Math. Soc. Bulgaria, Sil'nchev, 6-9 April 1988, Sophia (1988), pp. 147-150 M. van de Vel, "The fixed point property of superextensions," In: Gen. Topol. and Relat. Modern. Anal. and Algebra IV. Proc. 4th Prague Topoi. Symp., 1979, Part B, Prague (1977), pp. 477-480. M. van de Vel, "Dimension of convex hyperspaces: nonmetric case," Compos. Math., 50, No. 1, 95-108 (1983). M. van de Vel, "Dimension of convex hyperspaces," Fund. Math., 122, No. 2, 100-127 (1984). A. Verbeek, Superextensions of Topological Spaces, Math. Centre Tracts, No. 41, Amsterdam (1972). C.S. Vora, "Symmetric G-products of A-ANR," Publ. Ist. Math. Univ. Genova, 30 (1972). C.S. Vora, "Fixed points of symmetric product mappings of metric manifolds," Fund. Math., 85, No. 1, 19-24 (1974). C.S. Vora, "Fixed points of certain symmetric product mappings of a compact A-ANR," Math. Stud., 42, Nos. 1-4, 379396 (1974-1975). M.L. Wage, "The dimension of product spaces," Proc. Nat. Acad. Sci. USA, 75, No. 10, 4671-4672 (1978). C. Wagner, "Symmetric, cyclic, and permutation products on manifolds," Dissert. Math. Roz. Mat., 182, 3-48 (1980). L.E. Ward, Jr., "Extending Whitney maps," Pac. J. Math., 93, No. 2, 465-469 (1981). L.E. War, Jr., "Local selections and local dendrites," Topoi. Appl., 20, No. 1, 47-58 (1985). E. Wattel, "Superextensions embedded in cubes," Math. Centre. Tracts, No. 116, 305-317 (1979). E. Wattel, "A hereditarily separable compact ordered space X for which LX is not first countable," Math. Centre Tracts, No. 116, 319-321 (1979). J.E. West, "Infinite products which are Hilbert cubes," Trans. Am. Math. Soc., 81, No. 1, 163-165 (1975). M. Wejcicka, "Note on the Baire category in spaces of probability measures on nonseparable metrizable spaces," Bull. Pol. Acad. Sci. Math., 33, Nos. 5-6, 305-311 (1985). R . Y . T . Wong, "Some remarks on hyperspaces," Proc. Am. Math. Soc., 21, No. 3, 600-602 (1969). O. Wyler, "Algebraic theory of continuous lattices," Lect. Notes Math., 871, 390-413 (1981). Lin Xie, "Some results on the local compactness of hyperspaces," Acta Math. Sin., 26, No. 6, 650-656 (1983).
I7-5
406.
P. Zenor, "Hereditary ~ -separability and hereditary m-Lindel0f property in product spaces and function spaces," Fund. Math., 106, No. 3, 175-180 (1980). Additional Citations
407. 408. 409.
N.V. Velichko, "On a space of closed subsets," Sib. Mat. Zh., 16, No. 3, 627-629 (1975). V.I. Malykhin and B. I~. Shapirovskii, "The Martin axiom and the properties of.topological spaces," Dokl. Akad. Nauk SSSR, 213, No. 3, 532-535 (1973). J. Keesling, "On equivalence of normality and compactness in hyperspaces," Pac. J. Math., 33, No. 3, 657-667 (1870).
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
