Finitistic spaces and dimension (Q2716466)

The author studies finitistic spaces, defined as spaces for which every open cover has an open refinement of finite order. References are given to previous work on finitistic spaces done by S. Deo, J. Dydak, S. N. Mishra, A. R. Pears, R. A. Shukla, M. Singh, and H. S. Tripathi, as well as by the author. In this article he presents two main results. First, he finds a universal space for the class of metrizable finitistic spaces of a given weight. Second, he proves the following analogue of B. A. Pasynkov's dimension-theoretic factorization theorem: Let \(X\) be a normal space, \(Y\) a metrizable space, and \(f:X\to Y\) continuous. If \(X\) is a finitistic space, then there is a metrizable space \(Z\) and continuous mappings \(g:X\to Z\) and \(h:Z\to Y\) such that \(Z\) is a finitistic space, weight\((Z)\leq\text{weight}(Y)\), \(g(x)=Z\) and \(f=h\circ g\).