A selection theorem for topological convex structures

van de Vel, M.

doi:10.1090/S0002-9947-1993-1169083-9

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

A general theory of convexity

Tietze’s convexity theorem for semilattices and lattices

15

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

Images of

s

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

210

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

Collapsible polyhedra and median spaces

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

Modular normed spaces

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