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

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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 and any finitely homotopy dominated CW-complex , there exists a Hausdorff compactum with weight and which is universal for the property and weight . The condition means that for every closed subset of and every map , there exists a map which is an extension of . The universality means that for every compact Hausdorff space whose weight is and for which is true, there is an embedding of into .

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

**[Ch]**Alex Chigogidze,*Cohomological dimension of Tychonov spaces*, Topology Appl.**79**(1997), no. 3, 197–228. MR**1467214**, 10.1016/S0166-8641(96)00176-9**[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****[Dr]**A. N. Dranishnikov,*The Eilenberg-Borsuk theorem for mappings in an arbitrary complex*, Mat. Sb.**185**(1994), no. 4, 81–90 (Russian, with Russian summary); English transl., Russian Acad. Sci. Sb. Math.**81**(1995), no. 2, 467–475. MR**1272187**, 10.1070/SM1995v081n02ABEH003546**[DD]**A. Dranishnikov and J. Dydak,*Extension dimension and extension types*, Tr. Mat. Inst. Steklova**212**(1996), no. Otobrazh. i Razmer., 61–94; English transl., Proc. Steklov Inst. Math.**1 (212)**(1996), 55–88. MR**1635023****[Dy1]**Jerzy Dydak,*Cohomological dimension and metrizable spaces. II*, Trans. Amer. Math. Soc.**348**(1996), no. 4, 1647–1661. MR**1333390**, 10.1090/S0002-9947-96-01536-X**[Dy2]**J. Dydak,*Cohomological dimension theory*, Handbook of General Topology, Elsevier, Amsterdam, to appear.**[DM]**Jerzy Dydak and Jerzy Mogilski,*Universal cell-like maps*, Proc. Amer. Math. Soc.**122**(1994), no. 3, 943–948. MR**1242080**, 10.1090/S0002-9939-1994-1242080-7**[En1]**Ryszard Engelking,*General topology*, PWN—Polish Scientific Publishers, Warsaw, 1977. Translated from the Polish by the author; Monografie Matematyczne, Tom 60. [Mathematical Monographs, Vol. 60]. MR**0500780****[En2]**Ryszard Engelking,*Theory of dimensions finite and infinite*, Sigma Series in Pure Mathematics, vol. 10, Heldermann Verlag, Lemgo, 1995. MR**1363947****[Ma1]**J. T. Schwartz,*Another proof of E. Hopf’s ergodic lemma*, Comm. Pure Appl. Math.**12**(1959), 399–401. MR**0117542****[Ma2]**Sibe Mardešić,*Factorization theorems for cohomological dimension*, Topology Appl.**30**(1988), no. 3, 291–306. MR**972699**, 10.1016/0166-8641(88)90067-3**[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****[Pa]**B. A. Pasynkov,*A factorization theorem for the cohomological dimensions of mappings*, Vestnik Moskov. Univ. Ser. I Mat. Mekh.**4**(1991), 26–33, 103 (Russian, with Russian summary); English transl., Moscow Univ. Math. Bull.**46**(1991), no. 4, 26–31. MR**1205559****[Ru]**Leonard R. Rubin,*Cohomological dimension and approximate limits*, Proc. Amer. Math. Soc.**125**(1997), no. 10, 3125–3128. MR**1423333**, 10.1090/S0002-9939-97-04141-5**[RS]**Leonard R. Rubin and Philip J. Schapiro,*Compactifications which preserve cohomological dimension*, Glas. Mat. Ser. III**28(48)**(1993), no. 1, 155–165 (English, with English and Serbo-Croatian summaries). MR**1283905**

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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

Email:
mlevin@mozart.math.tulane.edu

**Leonard R. Rubin**

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

Email:
lrubin@ou.edu

**Philip J. Schapiro**

Affiliation:
Department of Mathematics, Langston University, Langston, Oklahoma 73050

Email:
pjschapiro@lunet.edu

DOI:
https://doi.org/10.1090/S0002-9939-00-05353-3

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