Decomposition spaces having arbitrarily small neighborhoods with -sphere boundaries

Author:
Edythe P. Woodruff

Journal:
Trans. Amer. Math. Soc. **232** (1977), 195-204

MSC:
Primary 57A10; Secondary 54B15

DOI:
https://doi.org/10.1090/S0002-9947-1977-0442944-2

MathSciNet review:
0442944

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let *G* be an u.s.c. decomposition of . Let *H* denote the set of nondegenerate elements and *P* be the natural projection of onto . Suppose that each point in the decomposition space has arbitrarily small neighborhoods with 2-sphere boundaries which miss . We prove in this paper that this condition implies that is homeomorphic to . This answers a question asked by Armentrout [1, p. 15]. Actually, the hypothesis concerning neighborhoods with 2-sphere boundaries is necessary only for the points of .

**[1]**Steve Armentrout,*Monotone decompositions of 𝐸³*, Topology Seminar (Wisconsin, 1965) Ann. of Math. Studies, No. 60, Princeton Univ. Press, Princeton, N.J., 1966, pp. 1–25. MR**0222865****[2]**Steve Armentrout,*A three-dimensional spheroidal space which is not a sphere*, Fund. Math.**68**(1970), 183–186. MR**0270350**, https://doi.org/10.4064/fm-68-2-183-186**[3]**W. T. Eaton,*The sum of solid spheres*, Michigan Math. J.**19**(1972), 193–207. MR**0309116****[4]**Robert D. Edwards and Leslie C. Glaser,*A method for shrinking decompositions of certain manifolds*, Trans. Amer. Math. Soc.**165**(1972), 45–56. MR**0295357**, https://doi.org/10.1090/S0002-9947-1972-0295357-6**[5]**O. G. Harrold Jr.,*A sufficient condition that a monotone image of the three-sphere be a topological three-sphere*, Proc. Amer. Math. Soc.**9**(1958), 846–850. MR**0103454**, https://doi.org/10.1090/S0002-9939-1958-0103454-9**[6]**N. Hosay, Erratum to*"The sum of a cube and a crumpled cube is*", Notices Amer. Math. Soc.**11**(1964), p. 152.**[7]**L. V. Keldyš,*Topological imbeddings in Euclidean space*, Proceedings of the Steklov Institute of Mathematics, No. 81 (1966). Translated from the Russian by J. Zilber, American Mathematical Society, Providence, R.I., 1968. MR**0232371****[8]**Lloyd L. Lininger,*Some results on crumpled cubes*, Trans. Amer. Math. Soc.**118**(1965), 534–549. MR**0178460**, https://doi.org/10.1090/S0002-9947-1965-0178460-7**[9]**A. Marin and Y. M. Visetti,*A general proof of Bing’s shrinkability criterion*, Proc. Amer. Math. Soc.**53**(1975), no. 2, 501–507. MR**0388319**, https://doi.org/10.1090/S0002-9939-1975-0388319-X**[10]**Louis F. McAuley,*Some upper semi-continuous decompositions of 𝐸³ into 𝐸³*, Ann. of Math. (2)**73**(1961), 437–457. MR**0126258**, https://doi.org/10.2307/1970312**[11]**-,*Upper semicontinuous decompositions of**into**and generalizations to metric spaces*, Topology of 3-Manifolds and Related Topics (Proc. Univ. of Georgia Inst., 1961), Prentice-Hall, Englewood Cliffs, N. J., 1962, pp. 21-26. MR**25**#4502.**[12]**T. M. Price,*A necessary condition that a cellular upper semi-continuous decomposition of 𝐸ⁿ yield 𝐸ⁿ*, Trans. Amer. Math. Soc.**122**(1966), 427–435. MR**0193627**, https://doi.org/10.1090/S0002-9947-1966-0193627-0**[13]**Myra J. Reed,*Decomposition spaces and separation properties*, Doctoral Dissertation, SUNY, Binghamton, 1971.**[14]**Gordon Thomas Whyburn,*Analytic Topology*, American Mathematical Society Colloquium Publications, v. 28, American Mathematical Society, New York, 1942. MR**0007095**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
57A10,
54B15

Retrieve articles in all journals with MSC: 57A10, 54B15

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1977-0442944-2

Article copyright:
© Copyright 1977
American Mathematical Society