Property and proper shape theory

Author:
R. B. Sher

Journal:
Trans. Amer. Math. Soc. **190** (1974), 345-356

MSC:
Primary 54C56

MathSciNet review:
0341389

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Abstract: A class of spaces called the spaces has arisen in the study of a possibly noncompact variant of cellularity. These spaces play a role in this new theory analogous to that of the spaces in cellularity theory. Herein it is shown that the locally compact metric space *X* is an space if and only if there exists a tree *T* such that *X* and *T* have the same proper shape. This result is then used to classify the proper shapes of the spaces, two such being shown to have the same proper shape if and only if their end-sets are homeomorphic. Also, a possibly noncompact analog of property , called , is defined and it is shown that if *X* is a closed connected subset of a piecewise linear *n*-manifold, then *X* is an space if and only if *X* is an space. Finally, it is shown that a locally finite connected simplicial complex is an space if and only if all of its homotopy and proper homotopy groups vanish.

**[1]**Steve Armentrout,*𝑈𝑉 properties of compact sets*, Trans. Amer. Math. Soc.**143**(1969), 487–498. MR**0273573**, 10.1090/S0002-9947-1969-0273573-7**[2]**B. J. Ball and R. B. Sher,*A theory of proper shape for locally compact metric spaces*, Bull. Amer. Math. Soc.**79**(1973), 1023–1026. MR**0334139**, 10.1090/S0002-9904-1973-13312-9**[3]**B. J. Ball and R. B. Sher,*A theory of proper shape for locally compact metric spaces*, Fund. Math.**86**(1974), 163–192. MR**0365472****[4]**Karol Borsuk,*Fundamental retracts and extensions of fundamental sequences*, Fund. Math. 64 (1969), 55-85; errata, ibid.**64**(1969), 375. MR**0243520****[5]**Edward M. Brown,*Proper homotopy theory in simplicial complexes*, Topology Conference (Virginia Polytech. Inst. and State Univ., Blacksburg, Va., 1973) Springer, Berlin, 1974, pp. 41–46. Lecture Notes in Math., Vol. 375. MR**0356041****[6]**Hans Freudenthal,*Neuaufbau der Endentheorie*, Ann. of Math. (2)**43**(1942), 261–279 (German). MR**0006504****[7]**Dean Hartley,*Quasi-cellularity in manifolds*, Ph.D. thesis, University of Georgia, Athens, Ga., 1973. (Condensed version, to appear).**[8]**J. R. Isbell,*Uniform spaces*, Mathematical Surveys, No. 12, American Mathematical Society, Providence, R.I., 1964. MR**0170323****[9]**W. Kuperberg,*Homotopically labile points of locally compact metric spaces*, Fund. Math.**73**(1971/72), no. 2, 133–136. MR**0298641****[10]**R. C. Lacher,*Cell-like spaces*, Proc. Amer. Math. Soc.**20**(1969), 598–602. MR**0234437**, 10.1090/S0002-9939-1969-0234437-3**[11]**D. R. McMillan Jr.,*A criterion for cellularity in a manifold*, Ann. of Math. (2)**79**(1964), 327–337. MR**0161320****[12]**-,*UV properties and related topics*, mimeographed lecture notes, Florida State University, Tallahassee, Fla., 1970.**[13]**Sibe Mardešić,*Retracts in shape theory*, Glasnik Mat. Ser. III**6(26)**(1971), 153–163 (English, with Serbo-Croatian summary). MR**0296915**

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DOI:
https://doi.org/10.1090/S0002-9947-1974-0341389-0

Keywords:
Property ,
property ,
cellularity,
quasi-cellularity,
proper shape,
proper homotopy type,
tree,
ends,
proper homotopical domination,
property ,
property ,
proper homotopy group

Article copyright:
© Copyright 1974
American Mathematical Society