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A total finite-dimensional selection theorem. (English. Russian original) Zbl 0954.54007
Sib. Math. J. 39, No. 5, 835-843 (1998); translation from Sib. Mat. Zh. 39, No. 5, 971-981 (1998).
A dim-filtration of a $$(p+1)$$-dimensional (in Hausdorff’s sense) paracompact space $$X$$ is a decreasing sequence $$X=X_p\supset X_{p-1}\supset\dots\supset X_{-1}\supset X_{-2}$$ of subspaces of $$X$$ such that $$\text{dim}_{X_t}(X_{t-1})\leq t$$ for all $$0\leq t\leq p$$. Assume that some equi-locally $$t$$-connected families $$\mathfrak S_t$$, $$-1\leq t\leq \infty$$, of closed subsets are fixed in a metric space $$(Z,\rho)$$. Assume also that $$\bigcup \mathfrak S_t$$ is closed in the union $$\bigcup\{\bigcup \mathfrak S_t\;\vrule\;0\leq t\leq\infty\}$$. The main result of the article is as follows:
Theorem A. Let $$\Phi\: X\to Z$$ be a lower semicontinuous multivalued mapping with complete (in the metric $$\rho$$) values $$\Phi (x)$$, $$x\in X$$, for which $$\Phi (x)\in \mathfrak S_t$$ whenever $$x\in X_t\smallsetminus X_{t-1}$$. Then, for every closed $$A\subset X$$ and every continuous selection $$r\:A\to L$$ of the restriction $$\Phi \vrule _A$$, there exist a neighborhood $$O(A)$$ of $$A$$ and a continuous extension $$r'\:O(A)\to Z$$ that is a selection of the mapping $$\Phi\vrule _{O(A)}$$. Moreover, if the families $$\mathfrak S_t$$ consist of $$t$$-connected sets then the neighborhood $$O(A)$$ may be assumed to equal $$X$$ (i.e., the local selection $$r$$ is extendible to some global selection).
If $$X=X_{-1}$$ then Theorem A coincides with the zero-dimensional selection theorem. If $$X=X_t$$ and $$X_{t-1}=\emptyset$$ then Theorem A coincides with the finite-dimensional selection theorem. If the set $$\{t\leq p\mid Y_t\neq\emptyset\}$$ consists of two elements then Theorem A is exactly Theorem 3 of [E. Michael, Pac. J. Math. 87, 189-197 (1980; Zbl 0405.54016)].
##### MSC:
 54C65 Selections in general topology 54F45 Dimension theory in general topology 54C20 Extension of maps
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##### References:
 [1] E. Michael, ”Continuous selections. II,” Ann. of Math. (2),64, No. 3, 562–580 (1956). · Zbl 0073.17702 [2] V. V. Fedorchuk, and V. V. Filippov, General Topology. Basic Constructions [in Russian], Moscow Univ., Moscow (1988). [3] E. Michael, ”Continuous selections and finite-dimensional sets,” Pacific J. Math.,87, 189–197 (1980). · Zbl 0435.54016 [4] S. T. Hu, Theory of Retracts, Wayne State Univ. Press, Detroit (1965). · Zbl 0145.43003 [5] C. Bessaga and A. Pelczynski, Selected Topics in Infinite-Dimensional Topology, Warszawa (1975). · Zbl 0304.57001 [6] D. Repovs and P. Semenov, A. Survey of E. Michael’s Theory of Continuous Selections and Its Applications [Preprint, No. 31], Lubljana (1993).
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