Shrinking countable decompositions of

Authors:
Richard Denman and Michael Starbird

Journal:
Trans. Amer. Math. Soc. **276** (1983), 743-756

MSC:
Primary 57N12; Secondary 54B15

MathSciNet review:
688975

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Abstract: Conditions are given which imply that a countable, cellular use decomposition is shrinkable. If the embedding of each element in has the bounded nesting property, defined in this paper, then is homeomorphic to . The bounded nesting property is a condition on the defining sequence of cells for an element of which implies that satisfies the Disjoint Disk criterion for shrinkability [ , Theorem 3.1]. From this result, one deduces that countable, star-like equivalent use decompositions of are shrinkable--a result proved independently by E. Woodruff [ ]. Also, one deduces the shrinkability of countable bird-like equivalent use decompositions (a generalization of the star-like result), and the recently proved theorem that if each element of a countable use decomposition of has a mapping cylinder neighborhood, then is shrinkable [ ; , Theorem 4.1; -, Theorem 1].

**[**R. H. Bing,**B1**]*Upper semicontinuous decompositions of 𝐸³*, Ann. of Math. (2)**65**(1957), 363–374. MR**0092960****[**R. H. Bing,**B2**]*Point-like decompositions of 𝐸³*, Fund. Math.**50**(1961/1962), 431–453. MR**0137104****[**Dan Everett,**E**]*Shrinking countable decompositions of 𝐸³ into points and tame cells*, Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), Academic Press, New York-London, 1979, pp. 53–72. MR**537724****[**Michael Starbird,**S1**]*Cell-like 0-dimensional decompositions of 𝐸³*, Trans. Amer. Math. Soc.**249**(1979), no. 1, 203–215. MR**526318**, 10.1090/S0002-9947-1979-0526318-3**[**Michael Starbird,**S2**]*Null sequence cellular decompositions of 𝑆³*, Fund. Math.**112**(1981), no. 2, 81–87. MR**619484****[**Michael Starbird and Edythe P. Woodruff,**S-W**]*Decompositions of 𝐸³ with countably many nondegenerate elements*, Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), Academic Press, New York-London, 1979, pp. 239–252. MR**537733****[**Edythe P. Woodruff,**W**]*Decompositions of**with countably many star-like equivalent elements*, preprint.

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1983-0688975-1

Keywords:
Countable cellular decomposition,
bounded nesting,
star-like,
bird-like

Article copyright:
© Copyright 1983
American Mathematical Society