The structure of nested spaces
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- by T. B. Muenzenberger and R. E. Smithson PDF
- Trans. Amer. Math. Soc. 201 (1975), 57-87 Request permission
Abstract:
The structure of nested spaces is studied in this paper using such tools as branches, chains, partial orders, and rays in the context of semitrees. A classification scheme for various kinds of acyclic spaces is delineated in terms of semitrees. Several families of order compatible topologies for semitrees are investigated, and these families are grouped in a spectrum (inclusion chain) of topologies compatible with the semitree structure. The chain, interval, and tree topologies are scrutinized in some detail. Several topological characterizations of semitrees with certain order compatible topologies are also derived.References
- Richard Arens, On the construction of linear homogeneous continua, Bol. Soc. Mat. Mexicana 2 (1945), 33–36. MR 0014390
- David P. Bellamy, Composants of Hausdorff indecomposable continua; a mapping approach, Pacific J. Math. 47 (1973), 303–309. MR 331345, DOI 10.2140/pjm.1973.47.303
- K. Borsuk, A theorem on fixed points, Bull. Acad. Polon. Sci. Cl. III. 2 (1954), 17–20. MR 0064393
- J. J. Charatonik, On ramification points in the classical sense, Fund. Math. 51 (1962/63), 229–252. MR 143183, DOI 10.4064/fm-51-3-229-252
- J. J. Charatonik and Carl Eberhart, On smooth dendroids, Fund. Math. 67 (1970), 297–322. MR 275372, DOI 10.4064/fm-67-3-297-322
- J. J. Charatonik and C. A. Eberhart, On contractible dendroids, Colloq. Math. 25 (1972), 89–98, 164. MR 309082, DOI 10.4064/cm-25-1-89-98
- James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606
- Orrin Frink Jr., Topology in lattices, Trans. Amer. Math. Soc. 51 (1942), 569–582. MR 6496, DOI 10.1090/S0002-9947-1942-0006496-X
- R. W. Hansell, Monotone subnets in partially ordered sets, Proc. Amer. Math. Soc. 18 (1967), 854–858. MR 215759, DOI 10.1090/S0002-9939-1967-0215759-7 J. K. Harris, Order structures for certain acyclic topological spaces, Thesis, University of Oregon, 1962.
- W. Holsztyński, Fixed points of arcwise connected spaces, Fund. Math. 64 (1969), 289–312. MR 248757, DOI 10.4064/fm-64-3-289-312
- Sibe Mardešić, A locally connected continuum which contains no proper locally connected subcontinuum, Glasnik Mat. Ser. III 2(22) (1967), 167–178 (English, with Serbo-Croatian summary). MR 221474 L. K. Mohler, Partial orders and the fixed point property for hereditarily unicoherent continua, Thesis, University of Oregon, 1968.
- L. Mohler, A fixed point theorem for continua which are hereditarily divisible by points, Fund. Math. 67 (1970), 345–358. MR 261583, DOI 10.4064/fm-67-3-345-358
- T. B. Muenzenberger and R. E. Smithson, Fixed point structures, Trans. Amer. Math. Soc. 184 (1973), 153–173 (1974). MR 328900, DOI 10.1090/S0002-9947-1973-0328900-X —, Mobs and semitrees, J. Austral. Math. Soc. (to appear).
- T. B. Muenzenberger and R. E. Smithson, Refluent multifunctions on semitrees, Proc. Amer. Math. Soc. 44 (1974), 189–195. MR 341462, DOI 10.1090/S0002-9939-1974-0341462-2
- T. Naito, On a problem of Wolk in interval topologies, Proc. Amer. Math. Soc. 11 (1960), 156–158. MR 110653, DOI 10.1090/S0002-9939-1960-0110653-8
- Robert L. Plunkett, A fixed point theorem for continuous multi-valued transformations, Proc. Amer. Math. Soc. 7 (1956), 160–163. MR 87094, DOI 10.1090/S0002-9939-1956-0087094-4
- Raymond E. Smithson, A note on acyclic continua, Colloq. Math. 19 (1968), 67–71. MR 226601, DOI 10.4064/cm-19-1-67-71
- Raymond E. Smithson, Topologies generated by relations, Bull. Austral. Math. Soc. 1 (1969), 297–306. MR 257956, DOI 10.1017/S0004972700042167
- Lynn A. Steen and J. Arthur Seebach Jr., Counterexamples in topology, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1970. MR 0266131
- L. B. Treybig, Concerning homogeneity in totally ordered, connected topological space, Pacific J. Math. 13 (1963), 1417–1421. MR 159309, DOI 10.2140/pjm.1963.13.1417
- 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., Mobs, trees, and fixed points, Proc. Amer. Math. Soc. 8 (1957), 798–804. MR 97036, DOI 10.1090/S0002-9939-1957-0097036-4
- L. E. Ward Jr., On dendritic sets, Duke Math. J. 25 (1958), 505–513. MR 98357
- L. E. Ward Jr., A fixed point theorem for multi-valued functions, Pacific J. Math. 8 (1958), 921–927. MR 103446, DOI 10.2140/pjm.1958.8.921
- L. E. Ward Jr., A fixed point theorem for chained spaces, Pacific J. Math. 9 (1959), 1273–1278. MR 108784, DOI 10.2140/pjm.1959.9.1273
- L. E. Ward Jr., Characterization of the fixed point property for a class of set-valued mappings, Fund. Math. 50 (1961/62), 159–164. MR 133122, DOI 10.4064/fm-50-2-159-164
- Lewis E. Ward Jr., Topology, Pure and Applied Mathematics, No. 10, Marcel Dekker, Inc., New York, 1972. An outline for a first course. MR 0467643
- Gordon Thomas Whyburn, Analytic Topology, American Mathematical Society Colloquium Publications, Vol. 28, American Mathematical Society, New York, 1942. MR 0007095, DOI 10.1090/coll/028
- E. S. Wolk, Order-compatible topologies on a partially ordered set, Proc. Amer. Math. Soc. 9 (1958), 524–529. MR 96596, DOI 10.1090/S0002-9939-1958-0096596-8
- E. S. Wolk, On partially ordered sets possessing a unique order-compatible topology, Proc. Amer. Math. Soc. 11 (1960), 487–492. MR 111706, DOI 10.1090/S0002-9939-1960-0111706-0
- Gail S. Young Jr., The introduction of local connectivity by change of topology, Amer. J. Math. 68 (1946), 479–494. MR 16663, DOI 10.2307/2371828
- G. S. Young, Fixed-point theorems for arcwise connected continua, Proc. Amer. Math. Soc. 11 (1960), 880–884. MR 117711, DOI 10.1090/S0002-9939-1960-0117711-2
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 201 (1975), 57-87
- MSC: Primary 54F05
- DOI: https://doi.org/10.1090/S0002-9947-1975-0355995-1
- MathSciNet review: 0355995