On finitistic spaces
A space \(X\) is said to be finitistic if every open cover of \(X\) has a finite dimensional open refinement. These spaces were popularized by G. E. Bredon who proved results about cohomological structure of fixed point sets of a \(G\)-space \(X\) where \(G\) is a topological group acting on a finitistic space \(X\). An open question raised by Bredon on the topology of finitistic spaces was solved in [\textit{S. Deo} and \textit{H. S. Tripathi}, Topology 21, 393-399 (1982; Zbl 0496.57013)] where it was shown that the orbit space of a finitistic space under a compact Lie group action is again finitistic. The converse of the above result was also proved in [\textit{S. Deo} and \textit{T. B. Singh}, J. Lond. Math. Soc., II. Ser. 25, 162-170 (1982; Zbl 0451.57019)]. The authors of the present paper have obtained a couple of nice characterizations of a paracompact finitistic space [one of these was also obtained earlier by \textit{Y. Hattori}, Quest. Answers Gen. Topology 3, 47-55 (1985; Zbl 0572.54019)] on the lines of topological dimension theory. They have used their characterizations in proving some other important results about finitistic spaces which, in turn, have further clarified the topology of finitistic spaces. Some connections of finitistic spaces with the cohomological dimension theory have also been discussed.
scientific article; zbMATH DE number 1367545