## The Mardesic factorization theorem for extension theory and c-separation

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- by Michael Levin, Leonard R. Rubin and Philip J. Schapiro
- Proc. Amer. Math. Soc.
**128**(2000), 3099-3106 - DOI: https://doi.org/10.1090/S0002-9939-00-05353-3
- Published electronically: April 28, 2000
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## Abstract:

We shall prove a type of Mardešić 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

- Alex Chigogidze,
*Cohomological dimension of Tychonov spaces*, Topology Appl.**79**(1997), no. 3, 197–228. MR**1467214**, DOI 10.1016/S0166-8641(96)00176-9 - 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** - 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**, DOI 10.1070/SM1995v081n02ABEH003546 - 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** - Jerzy Dydak,
*Cohomological dimension and metrizable spaces. II*, Trans. Amer. Math. Soc.**348**(1996), no. 4, 1647–1661. MR**1333390**, DOI 10.1090/S0002-9947-96-01536-X - J. Dydak,
*Cohomological dimension theory*, Handbook of General Topology, Elsevier, Amsterdam, to appear. - Jerzy Dydak and Jerzy Mogilski,
*Universal cell-like maps*, Proc. Amer. Math. Soc.**122**(1994), no. 3, 943–948. MR**1242080**, DOI 10.1090/S0002-9939-1994-1242080-7 - Ryszard Engelking,
*Topologia ogólna*, Biblioteka Matematyczna [Mathematics Library], vol. 47, Państwowe Wydawnictwo Naukowe (PWN), Warsaw, 1975 (Polish). MR**0500779** - Ryszard Engelking,
*Theory of dimensions finite and infinite*, Sigma Series in Pure Mathematics, vol. 10, Heldermann Verlag, Lemgo, 1995. MR**1363947** - J. T. Schwartz,
*Another proof of E. Hopf’s ergodic lemma*, Comm. Pure Appl. Math.**12**(1959), 399–401. MR**117542**, DOI 10.1002/cpa.3160120212 - Sibe Mardešić,
*Factorization theorems for cohomological dimension*, Topology Appl.**30**(1988), no. 3, 291–306. MR**972699**, DOI 10.1016/0166-8641(88)90067-3 - Keiô Nagami,
*Dimension theory*, Pure and Applied Mathematics, Vol. 37, Academic Press, New York-London, 1970. With an appendix by Yukihiro Kodama. MR**0271918** - 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** - Leonard R. Rubin,
*Cohomological dimension and approximate limits*, Proc. Amer. Math. Soc.**125**(1997), no. 10, 3125–3128. MR**1423333**, DOI 10.1090/S0002-9939-97-04141-5 - 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**

## Bibliographic 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
- 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
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**128**(2000), 3099-3106 - MSC (1991): Primary 54F45, 55M10
- DOI: https://doi.org/10.1090/S0002-9939-00-05353-3
- MathSciNet review: 1664406