Geodesic
Shape Equivalences of Whitney Continua of Curves
Canadian journal of mathematics, Tome 40 (1988) no. 1, pp. 217-227

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By a compactum, we mean a compact metric space. A continuum is a connected compactum. A curve is a 1-dimensional continuum. Let X be a continuum and let C(X) be the hyperspace of (nonempty) subcontinua of X, C(X) is metrized with the Hausdorff metric (e.g., see [12] or [18]). One of the most convenient tools in order to study the structure of C(X) is a monotone map ω:C(X) → [0, ω(X)] defined by H. Whitney [25]. A map ω:C(X) → [0, ω(X)] is said to be a Whitney map for C(X) provided that The continua {ω −1} (0 < t < ω(X)) are called the Whitney continua of X. We may think of the map ω as measuring the size of a continuum. Note that ω −1(0) is homeomorphic to X and ω −1(ω(X)) = {X}. Naturally, we are interested in the structures of ω −1(t)(0 < t< ω(X)). In [14], J. Krasinkiewicz proved that if X is a circle-like continuum and ω is any Whitney map for C(X), then for any 0 < t < ω(X)ω −1(t) is shape equivalent to X, i.e., Sh ω −1(t) = Sh X (e.g., see [1] or [17]). In [8], we proved the following: If one of the conditions (i) and (ii) is satisfied, then the shape morphism which is defined in [7] and [8], is a shape equivalence. 
Kato, Hisao. Shape Equivalences of Whitney Continua of Curves. Canadian journal of mathematics, Tome 40 (1988) no. 1, pp. 217-227. doi: 10.4153/CJM-1988-009-5
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[1] 1. Borsuk, K., Theory of shape, Monografie Matematyczne. 59 (Polish Scientific Publishers, Warszawa, 1975). Google Scholar

[2] 2. Case, J. H. and Chamberlin, R. E., Characterizations of tree-like continua, Pacific J. Math.. 76(1959), 73–84. Google Scholar

[3] 3. Čerin, Z. T., Homotopy properties of locally compact space at infinity — calmness and smoothness, Pacific J. Math.. 79 (1978), 69–91. Google Scholar

[4] 4. Čerin, Z. T. and Sostak, A. P., Some remarks on Borsuk's fundamental metric, Topology. I (North-Holland, New York, 1980), 233–252. Google Scholar

[5] 5. Goodykoontz, J. T. Jr. and Nadler, S. B. Jr., Whitney levels in hyperspaces of certain Peano continua, Trans. Amer. Math. Soc.. 274 (1982), 671–694. Google Scholar

[6] 6. Kato, H., Concerning hyperspaces of certain Peano continua and strong regularity of Whitney maps, Pacific J. Math.. 119 (1985), 159–167. Google Scholar

[7] 7. Kato, H., Shape properties of Whitney maps for hyperspaces, Trans. Amer. Math. Sot.. 297 (1986), 529–546. Google Scholar

[8] 8. Kato, H., Whitney continua of curves. Trans. Amer. Math. Soc, to appear. Google Scholar | DOI

[9] 9. Kato, H., Whitney continua of graphs admit all homotopy types of compact connected ANRs, Fund. Math., to appear. Google Scholar | DOI

[10] 10. Kato, H., Various types of Whitney maps on n-dimensional compact connected polyhedra (n ≧ 2), Topology and its Application. 97 (1986), 748–750. Google Scholar

[11] 11. Kato, H., Movability and homotopy, homology pro-groups of Whitney continua, J. Math. Soc. Japan. 39 (1987), 435–446. Google Scholar

[12] 12. Kelley, J. L., Hyperspaces of a continuum, Trans. Amer. Math. Soc.. 52 (1942), 22–36. Google Scholar

[13] 13. Krasinkiewicz, J., On the hyperspaces of snake-like and circle-like continua, Fund. Math.. 83 (1974), 155–164. Google Scholar

[14] 14. Krasinkiewicz, J., Shape properties of hyperspaces, Fund. Math.. 101 (1978), 79–91. Google Scholar

[15] 15. Krasinkiewicz, J. and Nadler, S. B. Jr, Whitney properties, Fund. Math.. 98 (1978), 165–180. Google Scholar

[16] 16. Lynch, M., Whitney levels in C(X) are absolute retracts, Proc. Amer. Math. Soc.. 97 (1986), 748–750. Google Scholar

[17] 17. Mardesic, S. and Segal, J., Shape theory (North-Holland Mathematical Library, 1982). Google Scholar

[18] 18. Nadler, S. B. Jr., Hyperspaces of sets, Pure and Appl. Math.. 49 (Dekker, New York, 1978). Google Scholar

[19] 19. Segal, J., Hyperspaces of the inverse limit space, Proc. Amer. Math. Soc.. 10 (1959), 706–709. Google Scholar

[20] 20. Rogersy, J. T. Jr., Applications of Vietoris-Begle theorem for multi-valued maps to the cohomology of hyperspaces, Michigan Math. J.. 22 (1975), 315–319. Google Scholar

[21] 21. Rogersy, J. T. Jr., The cone —hyperspace property, Can. J. Math.. 24 (1972), 279–285. Google Scholar

[22] 22. Rogersy, J. T. Jr., Whitney continua in the hyperspace C(X), Pacific J. Math.. 58 (1975), 569–584. Google Scholar

[23] 23. Spanier, E., Algebraic topology (McGraw-Hill, New York, 1966). Google Scholar

[24] 24. Ward, L. E. Jr., Extending Whitney maps, Pacific J. Math.. 93 (1981), 465–469. Google Scholar

[25] 25. Whitney, H., Regular families of curves I, Proc. Nat. Acad. Sci. U.S.A.. 18 (1932), 275–278. Google Scholar

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