Maps which preserve graphs
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- by Van C. Nall PDF
- Proc. Amer. Math. Soc. 101 (1987), 563-570 Request permission
Abstract:
In 1976 Eberhart, Fugate, and Gordh proved that the weakly confluent image of a graph is a graph. A much weaker condition on the map is introduced called partial confluence, and it is shown that the image of a graph is a graph if and only if the map is partially confluent. In addition, it is shown that certain properties of one-dimensional continua are preserved by partially confluent maps, generalizing theorems of Cook and Lelek, Tymchatyn and Lelek, and Grace and Vought. Also, some continua in addition to graphs are shown to be the images of partially confluent maps only.References
- C. A. Eberhart, J. B. Fugate, and G. R. Gordh Jr., Branchpoint covering theorems for confluent and weakly confluent maps, Proc. Amer. Math. Soc. 55 (1976), no. 2, 409–415. MR 410703, DOI 10.1090/S0002-9939-1976-0410703-7
- H. Cook and A. Lelek, Weakly confluent mappings and atriodic Suslinian curves, Canadian J. Math. 30 (1978), no. 1, 32–44. MR 481165, DOI 10.4153/CJM-1978-003-2
- M. K. Fort Jr., Images of plane continua, Amer. J. Math. 81 (1959), 541–546. MR 106441, DOI 10.2307/2372912
- E. E. Grace and Eldon J. Vought, Semi-confluent and weakly confluent images of tree-like and atriodic continua, Fund. Math. 101 (1978), no. 2, 151–158. MR 518350, DOI 10.4064/fm-101-2-151-158
- A. Lelek, On the topology of curves. II, Fund. Math. 70 (1971), no. 2, 131–138. MR 283765, DOI 10.4064/fm-70-2-131-138
- A. Lelek and E. D. Tymchatyn, Pseudo-confluent mappings and a classification of continua, Canadian J. Math. 27 (1975), no. 6, 1336–1348. MR 418022, DOI 10.4153/CJM-1975-136-4 S. Mazurkiewicz, Sur l’existence des continues indecomposables, Fund. Math. 25 (1935), 327-328.
- Van C. Nall, Weak confluence and $W$-sets, Proceedings of the 1983 topology conference (Houston, Tex., 1983), 1983, pp. 161–193. MR 738474
- James T. Rogers Jr., Orbits of higher-dimensional hereditarily indecomposable continua, Proc. Amer. Math. Soc. 95 (1985), no. 3, 483–486. MR 806092, DOI 10.1090/S0002-9939-1985-0806092-1
- Gordon Thomas Whyburn, Analytic topology, American Mathematical Society Colloquium Publications, Vol. XXVIII, American Mathematical Society, Providence, R.I., 1963. MR 0182943
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 101 (1987), 563-570
- MSC: Primary 54C10; Secondary 54F20, 54F50
- DOI: https://doi.org/10.1090/S0002-9939-1987-0908670-X
- MathSciNet review: 908670