A selection theorem for topological convex structures
HTML articles powered by AMS MathViewer
- by M. van de Vel PDF
- Trans. Amer. Math. Soc. 336 (1993), 463-496 Request permission
Abstract:
A continuous selection theorem has been obtained for multivalued functions, the values of which are convex sets of certain synthetic convex structures. Applications are given related with superextensions, (semi)lattices, spaces of order arcs, trees, Whitney levels in hyperspaces, and geometric topology. Applications to traditional convexity in vector spaces involve Beer’s approximation theorem and a fixed point theorem of Dugundji-Granas. Some other applications (a.o. an invariant arc theorem) appear elsewhere.References
- S. M. Aseev, Approximation of semicontinuous multivalued mappings by continuous ones, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 3, 460–476, 670 (Russian). MR 661142
- Gerald Beer, Approximate selections for upper semicontinuous convex valued multifunctions, J. Approx. Theory 39 (1983), no. 2, 172–184. MR 716928, DOI 10.1016/0021-9045(83)90090-4
- Edward G. Begle, A fixed point theorem, Ann. of Math. (2) 51 (1950), 544–550. MR 35433, DOI 10.2307/1969367
- Czesław Bessaga and Aleksander Pełczyński, Selected topics in infinite-dimensional topology, Monografie Matematyczne, Tom 58. [Mathematical Monographs, Vol. 58], PWN—Polish Scientific Publishers, Warsaw, 1975. MR 0478168
- Karol Borsuk, Theory of retracts, Monografie Matematyczne, Tom 44, Państwowe Wydawnictwo Naukowe, Warsaw, 1967. MR 0216473
- Richard C. O’Brien, On the openness of the barycentre map, Math. Ann. 223 (1976), no. 3, 207–212. MR 420221, DOI 10.1007/BF01360953
- Arrigo Cellina, A theorem on the approximation of compact multivalued mappings, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 47 (1969), 429–433 (1970) (English, with Italian summary). MR 276936
- Arrigo Cellina, A further result on the approximation of set-valued mappings, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 48 (1970), 412–416 (English, with Italian summary). MR 276935
- Haskell Cohen, A cohomological definition of dimension for locally compact Hausdorff spaces, Duke Math. J. 21 (1954), 209–224. MR 66637
- D. W. Curtis, Application of a selection theorem to hyperspace contractibility, Canad. J. Math. 37 (1985), no. 4, 747–759. MR 801425, DOI 10.4153/CJM-1985-040-7
- C. H. Dowker, On countably paracompact spaces, Canad. J. Math. 3 (1951), 219–224. MR 43446, DOI 10.4153/cjm-1951-026-2
- Pierre Duchet, Convexity in combinatorial structures, Proceedings of the 14th winter school on abstract analysis (Srní, 1986), 1987, pp. 261–293. MR 920860
- J. Dugundji, Absolute neighborhood retracts and local connectedness in arbitrary metric spaces, Compositio Math. 13 (1958), 229–246 (1958). MR 113217
- James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606
- James Dugundji and Andrzej Granas, Fixed point theory. I, Monografie Matematyczne [Mathematical Monographs], vol. 61, Państwowe Wydawnictwo Naukowe (PWN), Warsaw, 1982. MR 660439
- Carl Eberhart, Sam B. Nadler Jr., and William O. Nowell Jr., Spaces of order arcs in hyperspaces, Fund. Math. 112 (1981), no. 2, 111–120. MR 619487, DOI 10.4064/fm-112-2-111-120
- Jürgen Eckhoff, Radon’s theorem revisited, Contributions to geometry (Proc. Geom. Sympos., Siegen, 1978) Birkhäuser, Basel-Boston, Mass., 1979, pp. 164–185. MR 568498
- J. W. Ellis, A general set-separation theorem, Duke Math. J. 19 (1952), 417–421. MR 49268, DOI 10.1215/S0012-7094-52-01941-8
- Ryszard Engelking, Topologia ogólna, Państwowe Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna, Tom 47. [Mathematics Library. Vol. 47]. MR 0500779
- Gerhard Gierz, Karl Heinrich Hofmann, Klaus Keimel, Jimmie D. Lawson, Michael W. Mislove, and Dana S. Scott, A compendium of continuous lattices, Springer-Verlag, Berlin-New York, 1980. MR 614752, DOI 10.1007/978-3-642-67678-9
- R. E. Jamison, R. C. O’Brien, and P. D. Taylor, On embedding a compact convex set into a locally convex topological vector space, Pacific J. Math. 64 (1976), no. 1, 193–205. MR 425578, DOI 10.2140/pjm.1976.64.193 R. E. Jamison (R. E. Jamison-Waldner), A general theory of convexity, Dissertation, Univ. of Washington, Seattle, 1974. —, Tietze’s convexity theorem for semilattices and lattices, Semigroup Forum 15 (1978), 357-373.
- Shizuo Kakutani, Ein Beweis des Satzes von M. Eidelheit über konvexe Mengen, Proc. Imp. Acad. Tokyo 13 (1937), no. 4, 93–94 (German). MR 1568455
- Klaus Keimel and Andrzej Wieczorek, Kakutani property of the polytopes implies Kakutani property of the whole space, J. Math. Anal. Appl. 130 (1988), no. 1, 97–109. MR 926830, DOI 10.1016/0022-247X(88)90388-5
- John L. Kelley and Isaac Namioka, Linear topological spaces, Graduate Texts in Mathematics, No. 36, Springer-Verlag, New York-Heidelberg, 1976. With the collaboration of W. F. Donoghue, Jr., Kenneth R. Lucas, B. J. Pettis, Ebbe Thue Poulsen, G. Baley Price, Wendy Robertson, W. R. Scott, and Kennan T. Smith; Second corrected printing. MR 0394084 G. Kozlowski, Images of $ANR$’s, Trans. Amer. Math. Soc. (to appear).
- George Kozlowski, Jan van Mill, and John J. Walsh, AR-maps obtained from cell-like maps, Proc. Amer. Math. Soc. 82 (1981), no. 2, 299–302. MR 609671, DOI 10.1090/S0002-9939-1981-0609671-6
- Jimmie D. Lawson, Intrinsic topologies in topological lattices and semilattices, Pacific J. Math. 44 (1973), 593–602. MR 318031, DOI 10.2140/pjm.1973.44.593
- Jimmie D. Lawson, Embeddings of compact convex sets and locally compact cones, Pacific J. Math. 66 (1976), no. 2, 443–453. MR 440338, DOI 10.2140/pjm.1976.66.443
- Jean Leray, Théorie des points fixes: indice total et nombre de Lefschetz, Bull. Soc. Math. France 87 (1959), 221–233 (French). MR 143202, DOI 10.24033/bsmf.1519
- Ernest Michael, Continuous selections. I, Ann. of Math. (2) 63 (1956), 361–382. MR 77107, DOI 10.2307/1969615
- Ernest Michael, Continuous selections. II, Ann. of Math. (2) 64 (1956), 562–580. MR 80909, DOI 10.2307/1969603
- Ernest Michael, Continuous selections. III, Ann. of Math. (2) 65 (1957), 375–390. MR 83715, DOI 10.2307/1969969
- Ernest Michael, Convex structures and continuous selections, Canadian J. Math. 11 (1959), 556–575. MR 109344, DOI 10.4153/CJM-1959-051-9
- J. van Mill, Infinite-dimensional topology, North-Holland Mathematical Library, vol. 43, North-Holland Publishing Co., Amsterdam, 1989. Prerequisites and introduction. MR 977744
- J. van Mill and M. van de Vel, Convexity preserving mappings in subbase convexity theory, Nederl. Akad. Wetensch. Proc. Ser. A 40 (1978), no. 1, 76–90. MR 488568, DOI 10.1016/1385-7258(78)90025-2
- Jan van Mill and Marcel van de Vel, Subbases, convex sets, and hyperspaces, Pacific J. Math. 92 (1981), no. 2, 385–402. MR 618073, DOI 10.2140/pjm.1981.92.385
- Jan van Mill and Marcel van de Vel, Equality of the Lebesgue and the inductive dimension functions for compact spaces with a uniform convexity, Colloq. Math. 50 (1986), no. 2, 187–200. MR 857852, DOI 10.4064/cm-50-2-187-200
- Sam B. Nadler Jr., Hyperspaces of sets, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 49, Marcel Dekker, Inc., New York-Basel, 1978. A text with research questions. MR 0500811
- Lech Pasicki, On continuous selections, Opuscula Math. 3 (1987), 65–71 (1988) (English, with Russian and Polish summaries). MR 948598
- Ann Petrus, Contractibility of Whitney continua in $C(X)$, General Topology Appl. 9 (1978), no. 3, 275–288. MR 510909, DOI 10.1016/0016-660x(78)90031-4
- Carl P. Pixley, An example concerning continuous selections of infinite-dimensional spaces, Proc. Amer. Math. Soc. 43 (1974), 237–244. MR 328858, DOI 10.1090/S0002-9939-1974-0328858-X
- James W. Roberts, The embedding of compact convex sets in locally convex spaces, Canadian J. Math. 30 (1978), no. 3, 449–454. MR 470663, DOI 10.4153/CJM-1978-038-0
- R. E. Smithson, Fixed point theorems for certain classes of multifunctions, Proc. Amer. Math. Soc. 31 (1972), 595–600. MR 288750, DOI 10.1090/S0002-9939-1972-0288750-4
- Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
- Joseph L. Taylor, A counterexample in shape theory, Bull. Amer. Math. Soc. 81 (1975), 629–632. MR 375328, DOI 10.1090/S0002-9904-1975-13768-2
- John Tiller, Augmented compact spaces and continuous lattices, Houston J. Math. 7 (1981), no. 3, 441–453. MR 640985
- Hing Tong, Some characterizations of normal and perfectly normal spaces, Duke Math. J. 19 (1952), 289–292. MR 50265
- H. Toruńczyk, On $\textrm {CE}$-images of the Hilbert cube and characterization of $Q$-manifolds, Fund. Math. 106 (1980), no. 1, 31–40. MR 585543, DOI 10.4064/fm-106-1-31-40
- M. van de Vel, Finite-dimensional convex structures. I. General results, Topology Appl. 14 (1982), no. 2, 201–225. MR 667667, DOI 10.1016/0166-8641(82)90071-2
- M. van de Vel, Euclidean convexity cannot be compactified, Math. Ann. 262 (1983), no. 4, 563–572. MR 696526, DOI 10.1007/BF01456069 —, Pseudo-boundaries and pseudo-interiors for topological convexities, Dissertationes Math. 210 (1983), 1-72.
- M. van de Vel, Two-dimensional convexities are join-hull commutative, Topology Appl. 16 (1983), no. 2, 181–206. MR 712864, DOI 10.1016/0166-8641(83)90018-4
- M. van de Vel, Binary convexities and distributive lattices, Proc. London Math. Soc. (3) 48 (1984), no. 1, 1–33. MR 721770, DOI 10.1112/plms/s3-48.1.1
- M. van de Vel, Binary convexities and distributive lattices, Proc. London Math. Soc. (3) 48 (1984), no. 1, 1–33. MR 721770, DOI 10.1112/plms/s3-48.1.1
- M. van de Vel, Dimension of convex hyperspaces, Fund. Math. 122 (1984), no. 2, 107–127. MR 753019, DOI 10.4064/fm-122-2-107-127
- Marcel van de Vel, Lattices and semilattices: a convex point of view, Continuous lattices and their applications (Bremen, 1982) Lecture Notes in Pure and Appl. Math., vol. 101, Dekker, New York, 1985, pp. 279–302. MR 826008
- M. van de Vel, Convex Hilbert cubes in superextensions, Topology Appl. 22 (1986), no. 3, 255–266. MR 842659, DOI 10.1016/0166-8641(86)90024-6 —, Abstract, topological and uniform convex structures, Report WS-353, Monograph, Vrije Universiteit Amsterdam, 1989, 383 pp. —, Collapsible polyhedra and median spaces, Report WS-360, 1990.
- M. van de Vel, Invariant arcs, Whitney levels, and Kelley continua, Trans. Amer. Math. Soc. 326 (1991), no. 2, 749–771. MR 1010415, DOI 10.1090/S0002-9947-1991-1010415-8
- A. Verbeek, Superextensions of topological spaces, Mathematical Centre Tracts, No. 41, Mathematisch Centrum, Amsterdam, 1972. MR 0358698 E. Verheul, Modular normed spaces (submitted).
- L. E. Ward Jr., A note on dendrites and trees, Proc. Amer. Math. Soc. 5 (1954), 992–994. MR 71759, DOI 10.1090/S0002-9939-1954-0071759-2
- L. E. Ward Jr., A note on Whitney maps, Canad. Math. Bull. 23 (1980), no. 3, 373–374. MR 593400, DOI 10.4153/CMB-1980-055-4
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 336 (1993), 463-496
- MSC: Primary 46A99; Secondary 47H04, 54C65
- DOI: https://doi.org/10.1090/S0002-9947-1993-1169083-9
- MathSciNet review: 1169083