Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-22T13:13:49.786Z Has data issue: false hasContentIssue false
  • English
  • Français

Mutually Aposyndetic Decomposition of Homogeneous Continua

Dedicated to Charles L. Hagopian and James T. Rogers

Part of: Special properties Basic constructions

Published online by Cambridge University Press:  20 November 2018

Janusz R. Prajs
Janusz R. Prajs*
Affiliation:
California State University Sacramento, Department of Mathematics and Statistics, 6000 J Street, Sacramento, CA 95819, USA, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A new decomposition, the mutually aposyndetic decomposition of homogeneous continua into closed, homogeneous sets is introduced. This decomposition is respected by homeomorphisms and topologically unique. Its quotient is a mutually aposyndetic homogeneous continuum, and in all known examples, as well as in some general cases, the members of the decomposition are semi-indecomposable continua. As applications, we show that hereditarily decomposable homogeneous continua and path connected homogeneous continua are mutually aposyndetic. A class of new examples of homogeneous continua is defined. The mutually aposyndetic decomposition of each of these continua is non-trivial and different from Jones’ aposyndetic decomposition.

Keywords

ampleaposyndeticcontinuumdecompositionfilamenthomogeneous

MSC classification

Secondary: 54F15: Continua and generalizations 54B15: Quotient spaces, decompositions
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 62 , Issue 1 , 01 February 2010 , pp. 182 - 201
Copyright
Copyright © Canadian Mathematical Society 2010

References

[1] D. P., Bellamy, Short paths in homogeneous continua Topology Appl. 26(1987), 287-291. doi:10.1016/0166-8641()Google Scholar
[2] D. P., Bellamy and L., Lum, The cyclic connectivity of homogeneous arcwise connected continua. Trans. Amer. Math. Soc. 266(1981), 389-396. doi:10.2307/1998429Google Scholar
[3] D. P., Bellamy and J. M., Łysko, Connected open sets in products of indecomposable continua (preprint).Google Scholar
[4] R. H., Bing and F. B., Jones, Another homogeneous plane continuum Trans. Amer.Math. Soc. 90(1959), 171-192. doi:10.2307/1993272Google Scholar
[5] J. J., Charatonik and T. Maćkowiak, Around Effros’ theorem. Trans. Amer.Math. Soc. 298(1986), no 2, 579-602. doi:10.2307/2000637Google Scholar
[6] H. S., Davis, D. P., Stadtlander, and P. M., Swingle, Properties of the set functions Tn. Portugal. Math. 21(1962), 113-133.Google Scholar
[7] E. G., Effros, Transformation groups and C_-algebras. Ann. of Math. 81(1965), 38-55. doi:10.2307/1970381Google Scholar
[8] C. L., Hagopian, Mutual aposyndesis. Proc. Amer. Math. Soc. 23(1969), 615-622. doi:10.2307/2036598Google Scholar
[9] K. H., Hofmann and S. A., Morris, The Structure of Compact Groups. Second Revised and Augmented Edition. de Gruyter Studies in Mathematics 25. de Gruyter, Berlin, 2006.Google Scholar
[10] A., Illanes, Pairs of indecomposable continua whose product is mutually aposyndetic, Topology Proc. 22 (1997), 239-246.Google Scholar
[11] F. B., Jones, Aposyndetic continua and certain boundary problems. Amer. J.Math. 63(1941), 545-553. doi:10.2307/2371367Google Scholar
[12] F. B., Jones, On a certain type of homogeneous plane continuum. Proc. Amer. Math. Soc. 6(1955), 735-740. doi:10.2307/2032927Google Scholar
[13] J. L., Kelley, Hyperspaces of a continuum. Trans. Amer. Math. Soc. 52(1942), 22-36. doi:10.2307/1990151Google Scholar
[14] W., Lewis, Continuous curves of pseudo-arcs. Houston J. Math. 11(1985), no. 1, 91-99. doi:10.1016/0315-0860()90079-5Google Scholar
[15] Maćkowiak, T., Continuous mappings on continua. Dissertationes Math. 158(1979), 1-91.Google Scholar
[16] J. R., Prajs, A homogeneous arcwise connected non-locally-connected curve. Amer. J. Math. 124(2002), no. 4, 649-675. doi:10.1353/ajm.2002.0023Google Scholar
[17] J. R., Prajs, Mutual aposyndesis and products of solenoids. Topology Proc. 32(2008), 339-349.Google Scholar
[18] J. R., Prajs and Whittington, K., Filament sets and homogeneous continua. Topology Appl. 154(2007), no. 8, 1581-1591. doi:10.1016/j.topol.2006.12.005Google Scholar
[19] J. R., Prajs and Whittington, K., Filament additive homogeneous continua. Indiana Univ. Math. J. 56(2007), no. 1, 263-277. doi:10.1512/iumj.2007.56.2871Google Scholar
[20] J. R., Prajs and Whittington, K., Filament sets and decompositions of homogeneous continua. Topology Appl. 154(2007), no. 9, 1942-1956. doi:10.1016/j.topol.2007.01.012Google Scholar
[21] J. R., Prajs and Whittington, K., Filament sets, aposyndesis, and the decomposition theorem of Jones. Trans. Amer.Math. Soc. 359(2007), no. 12, 5991-6000. doi:10.1090/S0002-9947-07-04160-8Google Scholar
[22] J. T., Rogers, Jr., Completely regular mappings and homogeneous, aposyndetic continua. Canad. J. Math. 33(1981), no. 2, 450-453.Google Scholar
[23] J. T., Rogers, Cell-like decompositions of homogeneous continua. Proc. Amer.Math. Soc. 87(1983), no. 2, 375-377. doi:10.2307/2043720Google Scholar
[24] J. T., Rogers, Decompositions of continua over the hyperbolic plane. Trans. Amer.Math. Soc. 310(1988), no. 1, 277-291. doi:10.2307/2001121Google Scholar
[25] J. T., Rogers, Higher dimensional aposyndetic decompositions. Proc. Amer.Math. Soc. 131(2003), no. 10, 3285-3288. doi:10.1090/S0002-9939-03-06888-6Google Scholar
[26] G. S., Ungar, On all kinds of homogeneous spaces. Trans. Amer. Math. Soc. 212(1975), 393-400. doi:10.2307/1998635Google Scholar
[27] R. W., Wardle, On a property of J. L. Kelley. Houston J. Math. 3(1977), no. 2, 291-299.Google Scholar