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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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[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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