Spaces with a property related to uniformly local finiteness
A topological space X is said to have property (U) if every locally finite collection \({\mathcal F}\) of subsets of X is uniformly locally finite, that is, for \({\mathcal F}\) there is a normal open cover \({\mathcal U}\) of X such that each \(U\in {\mathcal U}\) intersects at most a finite number of members of \({\mathcal F}\). Every collectionwise normal and countably paracompact space is a space with property (U), and the latter is expandable. These notions and facts are due to \textit{K. Morita} [Surveys in general topology, 297-336 (1980; Zbl 0442.54025)]. In the present paper the notion of U-embedding is defined to describe spaces with property (U); a subspace A of a space X is said to be U-embedded (resp. \(U^{\aleph_ 0}\)-embedded) in X if every (resp. countable) uniformly locally finite collection of subsets of A is uniformly locally finite also in X. Results are (1) every closed subset of X is U-embedded in X iff every discrete collection of subsets of X is uniformly locally finite (such a space X is said to have property \((U^*))\), (2) a space has property (U) iff it has property \((U^*)\) and is a cb space, (3) A is \(U^{\aleph_ 0}\)- embedded in X iff \(A\times I\) is \(C_ 1\)-embedded in \(X\times I (I=[0,1])\) in the sense of \textit{T. Ishii} and \textit{H. Ohta} [Math. Jap. 23, 349-368 (1978; Zbl 0405.54010)], and other related ones.
collectionwise normal and countably paracompact space
uniformly locally finite collection of subsets
scientific article; zbMATH DE number 3864067