Arc-smooth continua

Fugate, J. B.; Gordh, G. R.; Lum, Lewis

doi:10.1090/S0002-9947-1981-0610965-3

Arc-smooth continua

Authors: J. B. Fugate, G. R. Gordh and Lewis Lum
Journal: Trans. Amer. Math. Soc. 265 (1981), 545-561
MSC: Primary 54F20; Secondary 54F50
DOI: https://doi.org/10.1090/S0002-9947-1981-0610965-3
MathSciNet review: 610965
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Continua admitting arc-structures and arc-smooth continua are introduced as higher dimensional analogues of dendroids and smooth dendroids, respectively. These continua include such spaces as: cones over compacta, convex continua in ${l_2}$ , strongly convex metric continua, injectively metrizable continua, as well as various topological semigroups, partially ordered spaces, and hyperspaces. The arc-smooth continua are shown to coincide with the freely contractible continua and with the metric -spaces of Stadtlander. Known characterizations of smoothness in dendroids involving closed partial orders, the set function , radially convex metrics, continuous selections, and order preserving mappings are extended to the setting of continua with arc-structures. Various consequences of the special contractibility properties of arc-smooth continua are also obtained.

[1] David P. Bellamy, An interesting plane dendroid, Fund. Math. 110 (1980), no. 3, 191–208. MR 602885, https://doi.org/10.4064/fm-110-3-191-207
[2] R. H. Bing, A convex metric with unique segments, Proc. Amer. Math. Soc. 4 (1953), 167–174. MR 0052763, https://doi.org/10.1090/S0002-9939-1953-0052763-6
[3] K. Borsuk, A theorem on fixed points, Bull. Acad. Polon. Sci. Cl. III. 2 (1954), 17–20. MR 0064393
[4] -, Theory of retracts, Polish Scientific Publishers, Warsaw, 1967.
[5] J. H. Carruth, Topics in quasi-ordered spaces, Dissertation, Louisiana State Univ., Baton Rouge, La., 1966.
[6] J. H. Carruth, A note on partially ordered compacta, Pacific J. Math. 24 (1968), 229–231. MR 0222852
[7] J. H. Case and R. E. Chamberlin, Characterizations of tree-like continua, Pacific J. Math. 10 (1960), 73–84. MR 0111000
[8] J. J. Charatonik and Carl Eberhart, On smooth dendroids, Fund. Math. 67 (1970), 297–322. MR 0275372, https://doi.org/10.4064/fm-67-3-297-322
[9] Carl Eberhart, Some classes of continua related to clan structures, Dissertation, Louisiana State Univ., Baton Rouge, La., 1966.
[10] Carl Eberhart, A note on smooth fans, Colloq. Math. 20 (1969), 89–90. MR 0243488, https://doi.org/10.4064/cm-20-1-89-90
[11] M. K. Fort Jr., Homogeneity of infinite products of manifolds with boundary, Pacific J. Math. 12 (1962), 879–884. MR 0145499
[12] J. B. Fugate, G. R. Gordh, Jr. and Lewis Lum, On arc-smooth continua, Topology Proc. 2 (1977), 645-656.
[13] G. R. Gordh Jr., Concerning closed quasi-orders on hereditarily unicoherent continua, Fund. Math. 78 (1973), no. 1, 61–73. MR 0322835, https://doi.org/10.4064/fm-78-1-61-73
[14] G. R. Gordh Jr. and Lewis Lum, Radially convex mappings and smoothness in continua, Houston J. Math. 4 (1978), no. 3, 335–342. MR 514247
[15] Charles L. Hagopian, A fixed point theorem for plane continua, Bull. Amer. Math. Soc. 77 (1971), 351–354. MR 0273591, https://doi.org/10.1090/S0002-9904-1971-12690-3
[16] John G. Hocking and Gail S. Young, Topology, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1961. MR 0125557
[17] J. R. Isbell, Six theorems about injective metric spaces, Comment. Math. Helv. 39 (1964), 65–76. MR 0182949, https://doi.org/10.1007/BF02566944
[18] F. Burton Jones, Concerning non-aposyndetic continua, Amer. J. Math. 70 (1948), 403–413. MR 0025161, https://doi.org/10.2307/2372339
[19] J. L. Kelley, Hyperspaces of a continuum, Trans. Amer. Math. Soc. 52 (1942), 22–36. MR 0006505, https://doi.org/10.1090/S0002-9947-1942-0006505-8
[20] Ronald J. Knill, Cones, products and fixed points, Fund. Math. 60 (1967), 35–46. MR 0211389, https://doi.org/10.4064/fm-60-1-35-46
[21] R. J. Koch and I. S. Krule, Weak cutpoint ordering on hereditarily unicoherent continua, Proc. Amer. Math. Soc. 11 (1960), 679–681. MR 0120606, https://doi.org/10.1090/S0002-9939-1960-0120606-1
[22] R. J. Koch and L. F. McAuley, Semigroups on continua ruled by arcs, Fund. Math. 56 (1964), 1–8. MR 0173240, https://doi.org/10.4064/fm-56-1-1-8
[23] J. Krasinkiewicz, No 0-dimensional set disconnects the hyperspace of a continuum, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 19 (1971), 755–758 (English, with Russian summary). MR 0307201
[24] Józef Krasinkiewicz and Piotr Minc, Dendroids and their endpoints, Fund. Math. 99 (1978), no. 3, 227–244. MR 0494010, https://doi.org/10.4064/fm-99-3-227-244
[25] A. Lelek, On plane dendroids and their end points in the classical sense, Fund. Math. 49 (1960/1961), 301–319. MR 0133806, https://doi.org/10.4064/fm-49-3-301-319
[26] Lewis Lum, A quasi order characterization of smooth continua, Pacific J. Math. 53 (1974), 495–500. MR 0358734
[27] Lewis Lum, A characterization of local connectivity in dendroids, Studies in topology (Proc. Conf., Univ. North Carolina, Charlotte, N.C., 1974; dedicated to Math. Sect. Polish Acad. Sci.), Academic Press, New York, 1975, pp. 331–338. MR 0358739
[28] Lewis Lum, Weakly smooth continua, Trans. Amer. Math. Soc. 214 (1975), 153-167. MR 0385820, https://doi.org/10.1090/S0002-9947-1975-0385820-4
[29] Lewis Lum, Order preserving and monotone retracts of a dendroid, Topology Proceedings, Vol. I (Conf., Auburn Univ., Auburn, Ala., 1976), Math. Dept., Auburn Univ., Auburn, Ala., 1977, pp. 57–61. MR 0458385
[30] T. Maćkowiak, Some characterizations of smooth continua, Fund. Math. 79 (1973), no. 2, 173–186. MR 0321033, https://doi.org/10.4064/fm-79-2-173-186
[31] T. Maćkowiak, On smooth continua, Fund. Math. 85 (1974), no. 1, 79–95. MR 0365532, https://doi.org/10.4064/fm-85-1-79-95
[32] L. Mohler, A characterization of smoothness in dendroids, Fund. Math. 67 (1970), 369–376. MR 0264619, https://doi.org/10.4064/fm-67-3-369-376
[33] Sam B. Nadler Jr., Hyperspaces of sets, Marcel Dekker, Inc., New York-Basel, 1978. A text with research questions; Monographs and Textbooks in Pure and Applied Mathematics, Vol. 49. MR 0500811
[34] Togo Nishiura and Choon Jai Rhee, The hyperspace of a pseudoarc is a Cantor manifold, Proc. Amer. Math. Soc. 31 (1972), 550–556. MR 0290337, https://doi.org/10.1090/S0002-9939-1972-0290337-4
[35] Raymond E. Smithson, A note on acyclic continua, Colloq. Math. 19 (1968), 67–71. MR 0226601, https://doi.org/10.4064/cm-19-1-67-71
[36] David Stadtlander, Thread actions, Duke Math. J. 35 (1968), 483–490. MR 0238297
[37] E. D. Tymchatyn, Some order theoretic characterizations of the 3-cell, Colloq. Math. 24 (1971/72), 195–203. MR 0303512, https://doi.org/10.4064/cm-24-2-195-203
[38] E. D. Tymchatyn, Partial order and collapsibility of 2-complexes, Fund. Math. 77 (1972), no. 1, 5–7. MR 0319177, https://doi.org/10.4064/fm-77-1-5-7
[39] L. E. Ward Jr., Mobs, trees, and fixed points, Proc. Amer. Math. Soc. 8 (1957), 798–804. MR 0097036, https://doi.org/10.1090/S0002-9939-1957-0097036-4
[40] L. E. Ward Jr., Rigid selections and smooth dendroids, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 19 (1971), 1041–1044 (English, with Russian summary). MR 0309079
[41] L. E. Ward Jr., Fixed point sets, Pacific J. Math. 47 (1973), 553–565. MR 0367963
[42] Gordon Thomas Whyburn, Analytic Topology, American Mathematical Society Colloquium Publications, v. 28, American Mathematical Society, New York, 1942. MR 0007095
[43] G. S. Young, Fixed-point theorems for arcwise connected continua, Proc. Amer. Math. Soc. 11 (1960), 880–884. MR 0117711, https://doi.org/10.1090/S0002-9939-1960-0117711-2