A general proof of Bing’s shrinkability criterion

Marin, A.; Visetti, Y. M.

doi:10.1090/S0002-9939-1975-0388319-X

A general proof of Bing’s shrinkability criterion
HTML articles powered by AMS MathViewer

by A. Marin and Y. M. Visetti PDF

Proc. Amer. Math. Soc. 53 (1975), 501-507 Request permission

Abstract:

This paper gives a proof of the general Bing shrinkability criterion, including a proof of the fundamental theorem that a shrinkable compact upper semicontinuous decomposition of a complete metric space is realized by a pseudo-isotopy of the space.

References

R. H. Bing, A homeomorphism between the $3$-sphere and the sum of two solid horned spheres, Ann. of Math. (2) 56 (1952), 354–362. MR 49549, DOI 10.2307/1969804
R. H. Bing, A decomposition of $E^3$ into points and tame arcs such that the decomposition space is topologically different from $E^3$, Ann. of Math. (2) 65 (1957), 484–500. MR 92961, DOI 10.2307/1970058
R. H. Bing, The cartesian product of a certain non-manifold and a line is $E_{4}$, Bull. Amer. Math. Soc. 64 (1958), 82–84. MR 97034, DOI 10.1090/S0002-9904-1958-10160-3
James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606
Robert D. Edwards and Leslie C. Glaser, A method for shrinking decompositions of certain manifolds, Trans. Amer. Math. Soc. 165 (1972), 45–56. MR 295357, DOI 10.1090/S0002-9947-1972-0295357-6
Louis F. McAuley, Some upper semi-continuous decompositions of $E^{3}$ into $E^{3}$, Ann. of Math. (2) 73 (1961), 437–457. MR 126258, DOI 10.2307/1970312

Upper semicontinuous decompositions of

into

and generalizations to metric spaces

25

Kiiti Morita and Sitiro Hanai, Closed mappings and metric spaces, Proc. Japan Acad. 32 (1956), 10–14. MR 87077
A. H. Stone, Metrizability of decomposition spaces, Proc. Amer. Math. Soc. 7 (1956), 690–700. MR 87078, DOI 10.1090/S0002-9939-1956-0087078-6
I. A. Vaiĭnšteĭn, On closed mappings of metric spaces, Doklady Akad. Nauk SSSR (N.S.) 57 (1947), 319–321 (Russian). MR 0022067