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)].

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)].

Reviewer: S.A.Malyugin (Novosibirsk)

##### MSC:

54C65 | Selections in general topology |

54F45 | Dimension theory in general topology |

54C20 | Extension of maps |

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\textit{S. M. Ageev} and \textit{D. Repovš}, Sib. Math. J. 39, No. 5, 971--981 (1998; Zbl 0954.54007); translation from Sib. Mat. Zh. 39, No. 5, 971--981 (1998)

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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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