CONTENTS Introduction......................................................................................5 1. General notion of aposyndesis....................................................6 2. Relation T for special families......................................................8 3. Properties of T.............................................................................9 4. T-aposyndesis in homogeneous continua..................................11 5. Colocal connectedness and T-aposyndesis...............................13 6. Decompositions and terminal continua......................................15 7. Decompositions of homogeneous continua...............................18 8. Indecomposable continua and colocal connectedness..............20 9. Closed domains in homogeneous continua...............................22 10. Irreducible continua.................................................................23 11. Decompositions onto locally connected continua.....................25 12. Locally connected homogeneous continua..............................29 13. Atriodic homogeneous continua..............................................30 14. Arcs and pseudoarcs in homogeneous continua.....................33 15. Homogeneous continua in compact 2-manifolds......................37 16. Multicoherence and homogeneity............................................41 17. Types of aposyndesis..............................................................42 18. More about decompositions.....................................................44 19. Mappings from hereditarily indecomposable continua.............46 20. Common model........................................................................48 21. Final remarks...........................................................................52 References....................................................................................54
University of Saskatchewan, Saskatoon, Saskatchewan, Canada
Bibliografia
[1] R. D. Anderson, One-dimensional continuous curves and a homogeneity theorem, Ann. of Math. 68 (1958), 1-16.
[2] D. P. Bellamy, Continua for which the set function T is continuous, Trans. Amer. Math. Soc. 151 (1970), 581-587.
[3] D. P. Bellamy, The cone over the Cantor set-continuous maps from both directions. Topology Conf., Emory Univ. Proc. 1970; 8-25.
[4] D. P. Bellamy, An uncountable collection of chainable continua, Trans. Amer. Math. Soc. 160 (1971), 297-304.
[5] D. P. Bellamy, Mappings of indecomposable continua, Proc. Amer. Math. Soc. 30 (1971), 179-180.
[6] D. P. Bellamy, Mapping hereditarily indecomposable continua onto a pseudo-arc, Lecture Notes in Math, Springer-Verlag 375, Top. Conf. Virginia 1973, 22-24.
[7] D. P. Bellamy, Composants of Hausdorff indecomposable continua, a mapping approach, Pacific J. Math. 47 (1973), 303-309.
[8] D. P. Bellamy, Continuous mappings between continua, Topology Conference, Guilford College, 1979, 101-111
[9] D. P. Bellamy, and H. S. Davis, Continuum neighbourhoods and filter bases, Proc. Amer. Math. Soc. 27 (1971), 371-374.
[10] D. P. Bellamy, and L. Lum, Simple closed curves in arcwise connected continua, Topology Conference Guilford College, 1979, 113-119.
[11] R. H. Bing, A homogeneous indecomposable plane continuum, Duke Math. J. 15 (1948), 729-742.
[12] R. H. Bing, Higher-dimensional hereditarily indecomposable continua, Trans. Amer. Math. Soc. 71 (1951), 267-273.
[13] R. H. Bing, Each homogeneous nondegenerate chainable continuum is a pseudo-arc, Proc. Amer. Math. Soc. 10 (1959), 345-346.
[14] R. H. Bing, A simple closed curve is the only homogeneous bounded plane continuum that contains an arc, Canad. J. Math. 12 (1960), 209-230.
[15] R. H. Bing and K. Borsuk, Some remarks concerning topologically homogeneous spaces, Ann. of Math. 81 (1965), 100-111.
[16] R. H. Bing and F. B. Jones, Another homogeneous plane continuum, Trans. Amer. Math. Soc. 90 (1959), 171-192.
[17] C. E. Burgess, Continua and various types of homogeneity, ibid. 88 (1958), 366-374.
[18] J.J. Charatonik, On decompositions of X-dendroids, Fund. Math. 67 (1970), 15-30.
[19] J.J. Charatonik, An example of a monostratic X-dendroid, ibid. 67 (1970), 75-87.
[20] J.J. Charatonik, Some problems concerning monotone decompositions of continua, Colloquia Math. Soc. J. Bolyai, Topics in Topology, Keszthely, 1972.
[21] J.J. Charatonik, On decompositions of continua. Fund. Math. 79 (1973), 113-130.
[23] J. Dydak and J. Segal, Shape theory, Springer Verlag, 1978.
[24] E.G. Effros, Transformation groups and C*-algebras, Ann. of Math. 81 (1965), 38-55.
[25] E. Emeryk and Z. Horbanowicz, On atomic mappings, Colloq. Math. 27 (1973), 49-55.
[26] L. Fearnley, Characterizations of the continuous images of the pseudoarc, Trans. Amer. Math. Soc. 111 (1964), 380-399.
[27] L. Fearnley, The pseudo-circle is not homogeneous, Bull. Amer. Math. Soc. 75 (1969), 554-558.
[28] L. Fearnley, Hereditarily indecomposable circularly chainable continua. Ph. D. Thesis, Univ. of London, 1970.
[29] R. W. Fitzgerald, Connected sets with a finite disconnection property, Studies in Topology, Academic Press, New York 1975, 139-173.
[30] R. W. Fitzgerald and P. M. Swingle, Core decompositions of continua, Fund. Math. 61 (1967), 33-50.
[31] M. K. Fort, Jr., Homogeneity of infinite products of manifolds with boundary, Pacific J. Math. 12 (1962), 879-884.
[32] G. R. Gordh, Jr., Monotone decompositions of irreducible Hausdorff continua, ibid. 36 (1971), 647-658.
[33] G. R. Gordh, Jr., On decompositions of smooth continua. Fund. Math. 75 (1972), 51-60.
[34] G. R. Gordh, Jr., Concerning closed quasi-orders on hereditarily unicoherent continua. Fund. Math. 78 (1973), 61-73.
[35] G. R. Gordh, Jr., On homogeneous hereditarily unicoherent continua, Proc. Amer. Math. Soc. 51 (1975), 198-202.
[36] E. E. Grace, A bibliography on aposyndesis, General Topology and Modern Analysis, Proc. Conf., Univ. California 1980, Academic Press, New York 1981, 493-513.
[37] E. E. Grace, Monotone decompositions of θ-continua, Trans. Amer. Math. Soc. 275 (1983), 275-295.
[38] E. E. Grace and E. J. Vought, Monotone decompositions of $θ_n$-continua, ibid. 263 (1981), 261-270.
[39] J. Grispolakis and E. D. Tymchatyn, A universal smooth dendroid. Bull. Acad. Polon. Sci. 12 (1978), 991-998.
[40] J. Grispolakis and E. D. Tymchatyn, Continua which admit only certain classes of onto mappings, Topology Proc. 3 (1978), 347-362.
[41] C. L. Hagopian, Homogeneous plane continua, Hauston J. Math. 1 (1975), 35-41.
[42] C. L. Hagopian, Indecomposable homogeneous plane continua are hereditarily indecomposable, Trans. Amer. Math. Soc. 224 (1976), 339-350.
[43] C. L. Hagopian, The fixed point property for almost chainable homogeneous continua, Illinois J. Math. 20 (1976), 650-652.
[44] C. L. Hagopian, A characterization of solenoids, Pacific J. Math. 68 (1977), 425-435.
[45] C. L. Hagopian and J. T. Rogers, Jr., A classification of homogeneous circle-like continua, Houston J. Math. 3 (1977), 471-474.
[46] W. Holsztyński, Une géneralization du théorème de Brouwer sur les points invariants, Bull. Acad. Polon. Sci. 12 (1964), 603-606.
[47] S. D. Iliadis, Universal continuum for the class of completely regular continua, ibid. 28 (1980), 603-607.
[48] W. T. Ingram, Concerning atriodic tree-like continua. Fund. Math. 101 (1978). 189-193.
[49] W. T. Ingram, Hereditarily indecomposable tree-like continua, ibid. 103 (1979), 61-64.
[50] Z. Janiszewski, Sur les continus irréductibles entre deux points, J. de l'Ecole Polytechnique (2) 16 (1912), 79-170.
[51] F. B. Jones, A note on homogeneous plane continua, Bull. Amer, Math. Soc. 55 (1949), 113-114.
[52] F. B. Jones, Certain homogeneous unicoherent indecomposable continua. Proc. Amer. Math. Soc. 2 (1951), 855-859.
[53] F. B. Jones, On a certain type of homogeneous plane continuum. Proc. Amer. Math. Soc. 6 (1955), 735-740.
[54] F. B. Jones, Homogeneous plane continua, Proc. Auburn Topol. Conf, Auburn University, 1969, 46 56.
[55] F. B. Jones, Use of a new technique in homogeneous continua. Houston J. Math. 1 (1975), 57 -61.
[56] V. L. Klee, Jr., Homogeneity of infinite-dimensional parallelotopes, Ann. of Math. 66 (1957), 454-460.
[57] B. Knaster, Sur les ensembles connexes irréductibles entre deux points, Fund. Math. 10 (1927), 276-297.
[58] B. Knaster, Un continu irreducible à décomposition continue en tranches, ibid. 25 (1935), 568-577.
[59] J. Krasinkiewicz, On two Dyer's theorems, Colloq. Math., to appear.
[65] J. Krasinkiewicz, Continua and their open subsets with connected complements, Fund. Math. 102 (1979), 129-136.
[66] W. Kuperberg, Uniformly pathwise connected continua, Studies in Topology, Proc, of the Charlotte Topology Conf. 1974, Academic Press, 1975, 315-324.
[67] W. Kuperberg and A. Lelek, Homotopy properties of pathwise connected continua, Fund. Math. 92 (1976), 29-41.
[68] K. Kuratowski, Théorie des continus irréductibles entre deux points I, ibid. 3 (1922), 200-231.
[69] K. Kuratowski, Théorie des continus irréductibles entre deux points II, ibid. 10 (1972), 225-275.
[70] K. Kuratowski, Topology, vol. II, Academic Press, New York and PWN, Warszawa 1968.
[71] A. Lelek, On weakly chainable continua, Fund. Math. 51 (1962), 271-282.
[72] A. Lelek, Disjoint mappings and the span of spaces, ibid. 55 (1964), 199-214.
[73] W. Lewis, Homogeneous hereditarily indecomposable continua-survey questions, Proc. of the 1980 Topology Conf., Alabama 1980, Top. Proc. 5 (1980), 215-222.
[74] W. Lewis, Monotone maps of hereditarily indecomposable continua, Proc. Amer. Math. Soc. 75
(1979), 361-364.
[75] W. Lewis and J. K. Phelps, Stable homeomorphisms, Galois spaces, and related properties in homogeneous continua, preprint.
[76] T. Maćkowiak, Planable and smooth dendroids, Proc. Fourth Prague Fourth Symp. (1976), 260-267.
[77] T. Maćkowiak, On some examples of monostratic λ-dendroids, Fund. Math. 87 (1975), 79-88.
[78] T. Maćkowiak, Continuous mappings on continua, Dissert. Math. 158 (1979), 1-91.
[79] T. Maćkowiak, On decompositions of hereditarily smooth continua, Fund. Math. 94 (1977), 25-33.
[80] T. Maćkowiak, Retracts of hereditarily unicoherent continua, Bull. Acad. Polon. Sci. ser. sci. math. 28
(1980), 177-183.
[81] T. Maćkowiak, Extension theorem for a pseudo-arc, Fund. Math., to appear.
[82] T. Maćkowiak and E. D. Tymchatyn, Some properties of open and related mappings, Colloq. Math., to appear.
[83] A. Mason and D. C. Wilson, Monotone mappings on n-dimensional continua, preprint.
[84] S. Mazurkiewicz, Sur les continus homogenes, Fund. Math. 5 (1924), 137-146.
[85] M. C. McCord, Universal p-like compacta, Michigan Math. J. 13 (1966), 71-85.
[86] M. C. McCord, Embedding of p-like compacta in manifolds, Canad. J. Math. 19 (1967), 331-332.
[87] J. Mioduszewski, A functional conception of snake-like continua, Fund. Math. 51 (1962), 179-189.
[88] E. E. Moise, A note on the pseudo-arc. Trans. Amer. Math. Soc. 64 (1949), 57-58.
[89] R. L. Moore, Triodic continua in the plane, Fund. Math. 13 (1929), 261-263.
[90] S. Nadler, Hyperspaces of sets, Pure and Applied Math. 49, M. Dekker, New York and Basel 1978.
[91] S. Nadler, Universal mappings and weakly conluent mappings, Fund. Math. 110 (1980), 221-235.
[92] G. Nöbeling, Regulär-eindimensionale Räume, Math. Ann. 104 (1931), 81-91.
[93] L. G. Oversteegen and E. D. Tymchatyn, Plane strips and the span of continua, Houston J. Math, to appear.
[94] Z. Rakowski, On decompositions of compact Hausdorff spaces. Bull. Acad. Polon. Sci. 23 (1975), 1089-1091.
[95] J. H. Roberts and N. E. Steenrod, Monotone transformations of two-dimensional manifolds, Ann. of Math. 39 (1938), 851-863.
[96] J. T. Rogers, Jr., Pseudo-circle and universal circularly chainable continua. Doctoral Dissert, Univ. of California, Riverside 1968.
[97] J. T. Rogers, Jr., The pseudo-circle is not homogeneous, Trans. Amer. Math. Soc. 148 (1970), 417-428.
[98] J. T. Rogers, Jr., Pseudo-circles and universal chainable continua, 111. J. Math. 14 (1970), 222-237.
[99] J. T. Rogers, Jr., Solenoids of pseudoarcs, Houston J. Math. 3 (1977), 531-537.
[100] J. T. Rogers, Jr., Completely regular mappings and homogeneous, aposyndetic continua, Canad. J. Math. 33 (1981), 450-453.
[101] J. T. Rogers, Jr., Homogeneous, separating plane continua are decomposable, Michigan J. Math., 28 (1981), 317-321.
[102] J. T. Rogers, Jr., Decompositions of homogeneous continua, Pacific J. Math, 99 (1982), 137-144.
[103] M.J. Russel, Monotone decompositions of continua irreducible about a finite set, Fund. Math. 72 (1971), 255-264.
[104] R. L. Russo, Universal continua, ibid. 105 (1979), 41-60.
[105] R. H. Sorgenfrey, Concerning triodic continua, Amer. J. Math. 66 (1944), 439-460.
[106] E. S. Thomas, Jr, Monotone decompositions of irreducible continua, Dissertat. Math. 50 (1966), 1-77.
[107] G. S. Ungar, On all kind of homogeneous spaces. Trans. Amer. Math. Soc. 212 (1975), 393-400.
[108] E.J. Vought, Monotone decompositions of continua not separated by any subcontinua, ibid. 192 (1974), 67-78.
[109] E.J. Vought, Monotone decompositions into trees of Hausdorff continua irreducible about a finite subset, Pacific J. Math. 54 (1974), 253-261.
[110] E.J. Vought, Monotone decompositions of Hausdorff continua, Proc. Amer. Math. Soc. 56 (1976), 371-376.
[111] Z. Waraszkiewicz, Une famille indenombrable de continus plans dont aucun n'est l'image continua d'un autre, Fund. Math. 18 (1932), 118-137.
[112] Z. Waraszkiewicz, Sur un problem de M. H. Hahn, ibid. 22 (1934), 180-205.
[113] T. Ważewski, Sur les courbes de Jordan ne refermant aucune courbe simple fermé de Jordan, Ann. Soc. Polon. Math. 2 (1923), 49-170.
[114] G. T. Whyburn, Analytic topology, AMS Colloq. Publ. 28, Providence, R.I., 1942.
[115] D. C. Wilson, Open mappings of the universal curve onto continuous curves. Trans. Amer. Math. Soc. 168 (1972), 497-515.
[116] H. C. Wisner, Decompositions and homogeneity of continua on a 2-manifold, Pacific J. Math. 12 (1962); 1145-1162.