Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Cohomological dimension and approximate limits


Author: Leonard R. Rubin
Journal: Proc. Amer. Math. Soc. 125 (1997), 3125-3128
MSC (1991): Primary 54F45, 55M10, 54B35
DOI: https://doi.org/10.1090/S0002-9939-97-04141-5
MathSciNet review: 1423333
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Approximate (inverse) systems of compacta have been useful in the study of covering dimension, dim, and cohomological dimension over an abelian group $G$, $\dim _{G}$. Such systems are more general than (classical) inverse systems. They have limits and structurally have similar properties. In particular, the limit of an approximate system of compacta satisfies the important property of being an approximate resolution. We shall prove herein that if $G$ is an abelian group, a compactum $X$ is the limit of an approximate system of compacta $X_{a}$, $n\in \mathbb {N}$, and $\dim _{G} X_{a}\leq n$ for each $a$, then $\dim _{G} X\leq n$.


References [Enhancements On Off] (What's this?)

  • [DD] A. Dranishnikov and J. Dydak, Extension theory of separable metrizable spaces with applications to dimension theory (preliminary version), preprint.
  • [DR] T. Dobrowolski and L. Rubin, The hyperspaces of infinite-dimensional compacta for covering and cohomological dimension are homeomorphic, Pacific J. of Math. 164 (1994), 15-39. MR 95a:54025
  • [Hu] S. Hu, Theory of Retracts, Wayne State University Press, Detroit, 1965. MR 31:6202
  • [Ma] S. Mardesic, Approximate inverse systems and resolutions, Fundamenta Math. 142 (1993), 241-255. MR 94f:54025
  • [MR1] S. Mardesic and L. Rubin, Approximate inverse systems of compacta and covering dimension, Pacific J. of Math. 138 (1989), 129-144. MR 90f:54058
  • [MR2] S. Mardesic and L. Rubin, Cell-like mappings and nonmetrizable compacta of finite cohomological dimension, Trans. Amer. Math. Soc. 313 (1989), 53-79. MR 90a:54095
  • [MRU] S. Mardesic, L. Rubin, and N. Uglesic, A note on approximate systems of metric compacta, Topology and its Appls. 59 (1994), 189-194. MR 96a:54008
  • [MS] S. Mardesic and J. Segal, Shape Theory, North-Holland, Amsterdam, 1982. MR 84b:55020
  • [MW] S. Mardesic and T. Watanabe, Approximate resolutions of spaces and mappings, Glasnik Mat. 24 (1989), 583-633. MR 92b:54019
  • [Na] K. Nagami, Dimension Theory, Academic Press, New York, 1970. MR 42:6799
  • [Wa] T. Watanabe, Numerical meshes and covering meshes of approximate inverse systems of compacta, Proc. Amer. Math. Soc. 123 (1995), 959-962. MR 95d:54007
  • [Wh] G. Whitehead, Elements of Homotopy Theory, Springer-Verlag, New York, 1978. MR 80b:55001

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 54F45, 55M10, 54B35

Retrieve articles in all journals with MSC (1991): 54F45, 55M10, 54B35


Additional Information

Leonard R. Rubin
Affiliation: Department of Mathematics, University of Oklahoma, 601 Elm Ave., Rm. 423, Norman, Oklahoma 73019
Email: LRUBIN@ou.edu

DOI: https://doi.org/10.1090/S0002-9939-97-04141-5
Keywords: Dimension, cohomological dimension, Eilenberg-Mac\, Lane complex, approximate (inverse) system, inverse system, resolution, approximate resolution
Received by editor(s): November 16, 1995
Communicated by: James West
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society