Abstract
We develop the method of approximative extension of mappings which enables us not only to simplify the proofs of many available theorems in the theory of extensors but also to obtain a series of new results. Combined with Ancel' theory of fiberwise trivial correspondences, this method leads to considerable progress in the characterization of absolute extensors in terms of local contractivity. We prove the following assertions: Suppose that a space X is represented as the union of countably many closed ANEM-subspaces X i and a countably dimensional subspace D: 1. If each X i is a strict deformation neighborhood retract of X and X∈LC, then X∈ANE. 2. If X∈LEC then X∈ANE.
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References
Hu S.-T., Theory of Retracts, Wayne State Univ. Press, Detroit (1965).
Haver W. E., “Locally contractible spaces that are absolute neighborhood retracts,” Proc. Amer. Math. Soc., 40, 280–284 (1973).
Toruńczyk H., “Concerning locally homotopy negligible sets and characterization of l2–manifolds,” Fund. Math., 101, 93–110 (1978).
Dowker C. H., “Homotopy extension theorems,” Proc. London Math. Soc., 6, 100–116 (1956).
Mardešič S., “Approximate polyhedra, resolutions of maps and shape fibrations,” Fund. Math., 114, 53–78 (1981).
Ancel F. D., “The role of countable dimensionality in the theory of cell-like relations,” Trans. Amer. Math. Soc., 287, 1–40 (1985).
Mardešič S., “Absolute neighborhood retracts and shape theory,” in: History of Topology, Elsevier Science, Amsterdam, 1999.
Cauty R., “Un espace metrique lineaire qui n'est pas un retracte absolu,” Fund. Math., 146, 85–99 (1994).
Nhu N. and Sakai K., “The compact neighborhood extension property and local equi-connectedness,” Proc. Amer. Math. Soc., 121, 259–265 (1994).
Borsuk K., Theory of Retracts, PWN, Warszawa (1967).
Addis D. F. and Gresham J. H., “A class of infinite dimensional spaces. Part 1: Dimension theory and Alexandroff's problem,” Fund. Math., 101, 195–205 (1978).
Aleksandrov P. S. and Pasynkov B. A., An Introduction to Dimension Theory [in Russian], Nauka, Moscow (1975).
Engelking R., General Topology [Russian translation], Mir, Moscow (1986).
Spanier E. N., Algebraic Topology [Russian translation], Mir, Moscow (1971).
Repovš D. and Semenov P., Continuous Selections of Multivalued Mappings, Kluwer, Dordrecht (1998). (Math. Appl.; 455.)
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Ageev, S.M., Repovš, D. The Method of Approximative Extension of Mappings in the Theory of Extensors. Siberian Mathematical Journal 43, 591–604 (2002). https://doi.org/10.1023/A:1016332916072
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DOI: https://doi.org/10.1023/A:1016332916072