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)



The Mardesic factorization theorem for extension theory and c-separation

Authors: Michael Levin, Leonard R. Rubin and Philip J. Schapiro
Journal: Proc. Amer. Math. Soc. 128 (2000), 3099-3106
MSC (1991): Primary 54F45, 55M10
Published electronically: April 28, 2000
MathSciNet review: 1664406
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We shall prove a type of Mardesic factorization theorem for extension theory over an arbitrary stratum of CW-complexes in the class of arbitrary compact Hausdorff spaces. Our result provides that the space through which the factorization occurs will have the same strong countability property (e.g., strong countable dimension) as the original one had. Taking into consideration the class of compact Hausdorff spaces, this result extends all previous ones of its type. Our factorization theorem will simultaneously include factorization for weak infinite-dimensionality and for Property C, that is, for C-spaces.

A corollary to our result will be that for any weight $\alpha $and any finitely homotopy dominated CW-complex $K$, there exists a Hausdorff compactum $X$ with weight $wX\leq \alpha $ and which is universal for the property $X\tau K$ and weight $\leq \alpha $. The condition $X\tau K$ means that for every closed subset $A$ of $X$ and every map $f:A\rightarrow K$, there exists a map $F:X\rightarrow K$which is an extension of $f$. The universality means that for every compact Hausdorff space $Y$ whose weight is $\leq \alpha $ and for which $Y\tau K$ is true, there is an embedding of $Y$ into $X$.

We shall show, on the other hand, that there exists a CW-complex $S$ which is not finitely homotopy dominated but which has the property that for each weight $\alpha $, there exists a Hausdorff compactum which is universal for the property $X\tau S$ and weight $\leq \alpha $.

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

  • [Ch] A. Chigogidze, Cohomological dimension of Tychonoff spaces, Topology and its Appls. 79 (1997), 197-228. MR 99f:55003
  • [DR] T. Dobrowolski and L. Rubin, The hyperspaces of infinite-dimensional compacta for covering and cohomological dimension are homeomorphic, Pacific Journal of Math. 164 (1994), 15-39. MR 95a:54025
  • [Dr] A Dranishnikov, The Eilenberg-Borsuk theorem for mappings into an arbitrary complex, Russian Acad. Sci. Sb. Math. 81 No. 2 (1995), 467-475. MR 95j:54028
  • [DD] A Dranishnikov and J. Dydak, Extension dimension and extension types, Trudy Mat. Inst. Steklov. 212 (1996), 61-94. MR 99h:54049
  • [Dy1] J. Dydak, Cohomological dimension and metrizable spaces. II, Trans. Amer. Math. Soc. 348 (1996), 1647-1661. MR 96h:55001
  • [Dy2] J. Dydak, Cohomological dimension theory, Handbook of General Topology, Elsevier, Amsterdam, to appear.
  • [DM] J. Dydak and J. Mogilski, Universal cell-like maps, Proc. Amer. Math. Soc. 122 (1994), 943-948. MR 95a:55003
  • [En1] R. Engelking, General Topology, PWN-Polish Scientific Publishers, Warsaw, 1977. MR 58:18316b
  • [En2] R. Engelking, Theory of Dimensions Finite and Infinite, Heldermann Verlag, Lemgo, Germany, 1995. MR 97j:54033
  • [Ma1] S. Mardesic, On covering dimension and inverse limits of compact spaces, Illinois J. of Math. 2 (1960), 278-291. MR 22:8320
  • [Ma2] S. Mardesic, Factorization theorems for cohomological dimension, Topology and its Appls. 30 (1988), 291-306. MR 90a:55004
  • [Na] K. Nagami, Dimension Theory, Academic Press, New York, 1970. MR 42:6799
  • [Pa] B. Pasynkov, A factorization theorem for cohomological dimensions of mappings, Moscow Univ. Math. Bull 46 (1991), 26-31. MR 94g:54024
  • [Ru] L. Rubin, Cohomological dimension and approximate limits, Proc. Amer. Math. Soc. 125 (1997), 3125-3128. MR 98g:55001
  • [RS] L. Rubin and P. Schapiro, Compactifications which preserve cohomological dimension, Glasnik Mat. 28(48) (1993), 155-165. MR 95g:54029

Similar Articles

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

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

Additional Information

Michael Levin
Affiliation: Department of Mathematics, Tulane University, New Orleans, Louisiana 70118
Address at time of publication: Department of Mathematics, Texas Tech University, Lubbock, Texas 79409

Leonard R. Rubin
Affiliation: Department of Mathematics, The University of Oklahoma, 601 Elm Avenue, Room 423, Norman, Oklahoma 73019

Philip J. Schapiro
Affiliation: Department of Mathematics, Langston University, Langston, Oklahoma 73050

Keywords: Cohomological dimension, covering dimension, extension theory, inverse sequences, inverse systems, inverse limits, compactification, Marde\v{s}i\'{c} factorization, weight, universal compactum, homotopy domination, C-space, weak infinite-dimension, strong countable-dimension
Received by editor(s): March 19, 1998
Received by editor(s) in revised form: November 13, 1998
Published electronically: April 28, 2000
Additional Notes: A portion of this work was completed while the first-named author was a J. Clarence Karcher Visitor in the Department of Mathematics at the University of Oklahoma.
Communicated by: Alan Dow
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society