The role of countable dimensionality in the theory of cell-like relations

Author:
Fredric D. Ancel

Journal:
Trans. Amer. Math. Soc. **287** (1985), 1-40

MSC:
Primary 54C55; Secondary 54C56, 54C60, 54F45

DOI:
https://doi.org/10.1090/S0002-9947-1985-0766204-X

MathSciNet review:
766204

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Consider only metrizable spaces. The notion of a slice-trivial relation is introduced, and Theorem 3.2 is proved. This theorem sets forth sufficient conditions for a continuous relation with compact point images to be slice-trivial. Theorem 4.5 posits a number of necessary and sufficient conditions for a map to be a hereditary shape equivalence. Several applications of these two theorems are made, including the following.

Theorem 5.1. *A cell-like map* *is a hereditary shape equivalence if there is a sequence* *of closed subsets of* *such that*

(1) *is countable dimensional, and*

(2) *is a hereditary shape equivalence for each* .

Theorem 5.9. *If* *is a proper onto map whose point inverses are* *sets, then* *is an absolute neighborhood extensor for the class of countable dimensional spaces. Furthermore, if* *is countable dimensional, then* *is an absolute neighborhood retract*.

Theorem 5.9 is of particular interest when specialized to the identity map of a locally contractible space.

**[**David F. Addis and John H. Gresham,**AG**]*A class of infinite-dimensional spaces. I. Dimension theory and Alexandroff’s problem*, Fund. Math.**101**(1978), no. 3, 195–205. MR**521122****[**F. D. Ancel,**A**]*The locally flat approximation of cell-like embedding relations*, Ph.D. Thesis, University of Wisconsin, Madison, 1976.**[**F. D. Ancel and J. W. Cannon,**AC**]*The locally flat approximation of cell-like embedding relations*, Ann. of Math. (2)**109**(1979), no. 1, 61–86. MR**519353**, https://doi.org/10.2307/1971267**[**Steve Armentrout and Thomas M. Price,**AP**]*Decompositions into compact sets with 𝑈𝑉 properties*, Trans. Amer. Math. Soc.**141**(1969), 433–442. MR**0244994**, https://doi.org/10.1090/S0002-9947-1969-0244994-3**[**J. W. Cannon,**C1**]*Taming cell-like embedding relations*, Geometric topology (Proc. Conf., Park City, Utah, 1974) Springer, Berlin, 1975, pp. 66–118. Lecture Notes in Math., Vol. 438. MR**0391104****[**J. W. Cannon,**C2**]*Taming codimension-one generalized submanifolds of 𝑆ⁿ*, Topology**16**(1977), no. 4, 323–334. MR**0500987**, https://doi.org/10.1016/0040-9383(77)90039-8**[**J. W. Cannon,**C3**]*Σ²𝐻³=𝑆⁵/𝐺*, Rocky Mountain J. Math.**8**(1978), no. 3, 527–532. MR**0478166**, https://doi.org/10.1216/RMJ-1978-8-3-527**[**J. W. Cannon,**C4**]*Shrinking cell-like decompositions of manifolds. Codimension three*, Ann. of Math. (2)**110**(1979), no. 1, 83–112. MR**541330**, https://doi.org/10.2307/1971245**[**T. A. Chapman,**Ch**]*Lectures on Hilbert cube manifolds*, American Mathematical Society, Providence, R. I., 1976. Expository lectures from the CBMS Regional Conference held at Guilford College, October 11-15, 1975; Regional Conference Series in Mathematics, No. 28. MR**0423357****[**R. J. Daverman and J. J. Walsh,**DW**]*Examples of cell-like maps that are not shape equivalences*, Michigan Math. J.**30**(1983), no. 1, 17–30. MR**694925**, https://doi.org/10.1307/mmj/1029002784**[**James Dugundji,**D**]*Topology*, Allyn and Bacon, Inc., Boston, Mass., 1966. MR**0193606****[**R. D. Edwards,**E**]*The double suspension of certain homology*-*spheres is*, Notices Amer. Math. Soc.**22**(1975), A-334; Abstract #757-G33.**[**Daniel L. Everett,**Ev**]*Embedding theorems for decomposition spaces*, Houston J. Math.**3**(1977), no. 3, 351–368. MR**0464241****[**Michael Hartley Freedman,**F**]*The topology of four-dimensional manifolds*, J. Differential Geom.**17**(1982), no. 3, 357–453. MR**679066****[**John H. Gresham,**G**]*A class of infinite-dimensional spaces. II. An extension theorem and the theory of retracts*, Fund. Math.**107**(1980), no. 3, 237–245. MR**585553****[**William E. Haver,**H1**]*Mappings between 𝐴𝑁𝑅s that are fine homotopy equivalences*, Pacific J. Math.**58**(1975), no. 2, 457–461. MR**0385865****[**William E. Haver,**H2**]*A covering property for metric spaces*, Topology Conference (Virginia Polytech. Inst. and State Univ., Blacksburg, Va., 1973) Springer, Berlin, 1974, pp. 108–113. Lecture Notes in Math., Vol. 375. MR**0365504****[**William E. Haver,**H3**]*Locally contractible spaces that are absolute neighborhood retracts*, Proc. Amer. Math. Soc.**40**(1973), 280–284. MR**0331311**, https://doi.org/10.1090/S0002-9939-1973-0331311-X**[**William E. Haver,**H4**]*A near-selection theorem*, General Topology Appl.**9**(1978), no. 2, 117–124. MR**503224****[**Sze-tsen Hu,**Hu**]*Theory of retracts*, Wayne State University Press, Detroit, 1965. MR**0181977****[**George Kozlowski,**K1**]*Factorization of certain maps up to homotopy*, Proc. Amer. Math. Soc.**21**(1969), 88–92. MR**0238312**, https://doi.org/10.1090/S0002-9939-1969-0238312-X**[**-,**K2**]*Images of*, Trans. Amer. Math. Soc. (to appear).**[**K. Kuratowski,**Ku**]*Topology*. II, Academic Press, New York, 1968.**[**R. C. Lacher,**L1**]*Cell-like spaces*, Proc. Amer. Math. Soc.**20**(1969), 598–602. MR**0234437**, https://doi.org/10.1090/S0002-9939-1969-0234437-3**[**R. C. Lacher,**L2**]*Cell-like mappings. I*, Pacific J. Math.**30**(1969), 717–731. MR**0251714****[**D. R. McMillan Jr.,**M**]*A criterion for cellularity in a manifold*, Ann. of Math. (2)**79**(1964), 327–337. MR**0161320**, https://doi.org/10.2307/1970548**[**Jun-iti Nagata,**N**]*Modern dimension theory*, Bibliotheca Mathematica, Vol. VI. Edited with the cooperation of the “Mathematisch Centrum” and the “Wiskundig Genootschap” at Amsterdam, Interscience Publishers John Wiley & Sons, Inc., New York, 1965. MR**0208571****[**Roman Pol,**P**]*A weakly infinite-dimensional compactum which is not countable-dimensional*, Proc. Amer. Math. Soc.**82**(1981), no. 4, 634–636. MR**614892**, https://doi.org/10.1090/S0002-9939-1981-0614892-2**[**Stephen Smale,**Sm**]*A Vietoris mapping theorem for homotopy*, Proc. Amer. Math. Soc.**8**(1957), 604–610. MR**0087106**, https://doi.org/10.1090/S0002-9939-1957-0087106-9**[**Edwin H. Spanier,**Sp**]*Algebraic topology*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0210112****[**Lynn Arthur Steen and J. Arthur Seebach Jr.,**SS**]*Counterexamples in topology*, 2nd ed., Springer-Verlag, New York-Heidelberg, 1978. MR**507446****[**Joseph L. Taylor,**T**]*A counterexample in shape theory*, Bull. Amer. Math. Soc.**81**(1975), 629–632. MR**0375328**, https://doi.org/10.1090/S0002-9904-1975-13768-2**[**H. Toruńczyk,**To**]*On 𝐶𝐸-images of the Hilbert cube and characterization of 𝑄-manifolds*, Fund. Math.**106**(1980), no. 1, 31–40. MR**585543****[**J. H. C. Whitehead,**W**]*A certain open manifold whose group is unity*, Quart. J. Math.**6**(1935), 268-279.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
54C55,
54C56,
54C60,
54F45

Retrieve articles in all journals with MSC: 54C55, 54C56, 54C60, 54F45

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1985-0766204-X

Article copyright:
© Copyright 1985
American Mathematical Society