Cohomological dimension and approximate limits
HTML articles powered by AMS MathViewer
- by Leonard R. Rubin PDF
- Proc. Amer. Math. Soc. 125 (1997), 3125-3128 Request permission
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
- A. Dranishnikov and J. Dydak, Extension theory of separable metrizable spaces with applications to dimension theory (preliminary version), preprint.
- Tadeusz Dobrowolski and Leonard R. Rubin, The hyperspaces of infinite-dimensional compacta for covering and cohomological dimension are homeomorphic, Pacific J. Math. 164 (1994), no. 1, 15–39. MR 1267500
- Sze-tsen Hu, Theory of retracts, Wayne State University Press, Detroit, 1965. MR 0181977
- S. Mardešić, On approximate inverse systems and resolutions, Fund. Math. 142 (1993), no. 3, 241–255. MR 1220551, DOI 10.4064/fm-142-3-241-255
- Sibe Mardešić and Leonard R. Rubin, Approximate inverse systems of compacta and covering dimension, Pacific J. Math. 138 (1989), no. 1, 129–144. MR 992178
- Sibe Mardešić and Leonard R. Rubin, Cell-like mappings and nonmetrizable compacta of finite cohomological dimension, Trans. Amer. Math. Soc. 313 (1989), no. 1, 53–79. MR 962284, DOI 10.1090/S0002-9947-1989-0962284-0
- S. Mardešić, L. Rubin, and N. Uglešić, A note on approximate systems of metric compacta, Topology Appl. 59 (1994), no. 2, 189–194. MR 1296032, DOI 10.1016/0166-8641(94)90094-9
- Sibe Mardešić and Jack Segal, Shape theory, North-Holland Mathematical Library, vol. 26, North-Holland Publishing Co., Amsterdam-New York, 1982. The inverse system approach. MR 676973
- S. Mardešić and T. Watanabe, Approximate resolutions of spaces and mappings, Glas. Mat. Ser. III 24(44) (1989), no. 4, 587–637 (English, with Serbo-Croatian summary). MR 1080085
- Keiô Nagami, Dimension theory, Pure and Applied Mathematics, Vol. 37, Academic Press, New York-London, 1970. With an appendix by Yukihiro Kodama. MR 0271918
- Tadashi Watanabe, Numerical meshes and covering meshes of approximate inverse systems of compacta, Proc. Amer. Math. Soc. 123 (1995), no. 3, 959–962. MR 1254858, DOI 10.1090/S0002-9939-1995-1254858-5
- George W. Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York-Berlin, 1978. MR 516508
Additional Information
- Leonard R. Rubin
- Affiliation: Department of Mathematics, University of Oklahoma, 601 Elm Ave., Rm. 423, Norman, Oklahoma 73019
- Email: LRUBIN@ou.edu
- Received by editor(s): November 16, 1995
- Communicated by: James West
- © Copyright 1997 American Mathematical Society
- 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