Cohomological dimension and approximate limits

Author:
Leonard R. Rubin

Journal:
Proc. Amer. Math. Soc. **125** (1997), 3125-3128

MSC (1991):
Primary 54F45, 55M10, 54B35

MathSciNet review:
1423333

Full-text PDF Free Access

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 , . 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 is an abelian group, a compactum is the limit of an approximate system of compacta , , and for each , then .

**[DD]**A. Dranishnikov and J. Dydak,*Extension theory of separable metrizable spaces with applications to dimension theory (preliminary version)*, preprint.**[DR]**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****[Hu]**Sze-tsen Hu,*Theory of retracts*, Wayne State University Press, Detroit, 1965. MR**0181977****[Ma]**S. Mardešić,*On approximate inverse systems and resolutions*, Fund. Math.**142**(1993), no. 3, 241–255. MR**1220551****[MR1]**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****[MR2]**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**, 10.1090/S0002-9947-1989-0962284-0**[MRU]**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**, 10.1016/0166-8641(94)90094-9**[MS]**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****[MW]**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****[Na]**Keiô Nagami,*Dimension theory*, With an appendix by Yukihiro Kodama. Pure and Applied Mathematics, Vol. 37, Academic Press, New York-London, 1970. MR**0271918****[Wa]**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**, 10.1090/S0002-9939-1995-1254858-5**[Wh]**George W. Whitehead,*Elements of homotopy theory*, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York-Berlin, 1978. MR**516508**

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:
http://dx.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