A characterization of locally connected continua by hyperspace retractions

Nadler, Sam B.

doi:10.1090/S0002-9939-1977-0458378-6

A characterization of locally connected continua by hyperspace retractions
HTML articles powered by AMS MathViewer

by Sam B. Nadler PDF

Proc. Amer. Math. Soc. 67 (1977), 167-176 Request permission

Abstract:

Let X be a (metric) continuum. It is shown that X is locally connected if and only if there is special type of retraction from ${2^X}$ onto $C(X)$, where ${2^X}$ [resp., $C(X)$] is the space of all nonempty compact subsets [resp., subcontinua] of X with the Hausdorff metric. Also, necessary and sufficient conditions are given for the “continuity of balls".

References

Nicolas Artémiadis, A remark on metric spaces, Nederl. akad. Wetensch. Proc. Ser. A 68=Indag. Math. 27 (1965), 316–318. MR 0185574
R. H. Bing, Partitioning a set, Bull. Amer. Math. Soc. 55 (1949), 1101–1110. MR 35429, DOI 10.1090/S0002-9904-1949-09334-5
Jack T. Goodykoontz Jr., Connectedness im kleinen and local connectedness in $2^{X}$ and $C(X)$, Pacific J. Math. 53 (1974), 387–397. MR 385782
Jack T. Goodykoontz Jr., Some functions on hyperspaces of hereditarily unicoherent continua, Fund. Math. 95 (1977), no. 1, 1–10. MR 436097, DOI 10.4064/fm-95-1-1-10
J. L. Kelley, Hyperspaces of a continuum, Trans. Amer. Math. Soc. 52 (1942), 22–36. MR 6505, DOI 10.1090/S0002-9947-1942-0006505-8
Robert B. Fraser Jr., A new characterization of Peano continua, Comment. Math. Prace Mat. 16 (1972), 247–248. MR 322826
K. Kuratowski, S. B. Nadler Jr., and G. S. Young, Continuous selections on locally compact separable metric spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 18 (1970), 5–11 (English, with Russian summary). MR 264630
J. D. Lawson, Applications of topological algebra to hyperspace problems, Topology (Proc. Ninth Annual Spring Topology Conf., Memphis State Univ., Memphis, Tenn., 1975) Lecture Notes in Pure and Appl. Math., Vol. 24, Dekker, New York, 1976, pp. 201–206. MR 0428252
Karl Menger, Untersuchungen über allgemeine Metrik, Math. Ann. 100 (1928), no. 1, 75–163 (German). MR 1512479, DOI 10.1007/BF01448840
Edwin E. Moise, Grille decomposition and convexification theorems for compact metric locally connected continua, Bull. Amer. Math. Soc. 55 (1949), 1111–1121. MR 35430, DOI 10.1090/S0002-9904-1949-09336-9
Sam B. Nadler Jr., Some problems concerning hyperspaces, Topology Conference (Virginia Polytech. Inst. and State Univ., Blacksburg, Va., 1973) Lecture Notes in Math., Vol. 375, Springer, Berlin, 1974, pp. 190–197. MR 0370465
Jack Segal, Hyperspaces of the inverse limit space, Proc. Amer. Math. Soc. 10 (1959), 706–709. MR 108780, DOI 10.1090/S0002-9939-1959-0108780-6
Gordon Thomas Whyburn, Analytic Topology, American Mathematical Society Colloquium Publications, Vol. 28, American Mathematical Society, New York, 1942. MR 0007095

Rétractes absolus et hyperespaces des continus

32