An approximate resolution of the product with a compact factor

Abstract

For any given approximate resolution $p=\{p_{a}|a\in A\}:X \rightarrow \mathscr{X}=(X_{a}, \mathscr{U}_{a}, p_{aa^{\prime}}, A)$ of a topological space $X$, where $\mathscr{X}$ is uniform, all $X_{a}$ are paracompact, all $\mathscr{U}_{a}$ are locally finite and $A$ is cofinite, and any given compact Hausdorff space $Y$, the approximate resolution $r=p\times 1=\{r_{b}=p_{a}\times 1|b=(a, \varphi)\in B\}$: $X \times Y \rightarrow \mathscr{X} \times Y =(X_{a}\times Y, \mathscr{U}_{a}\times\varphi[\mathscr{U}_{a}], p_{aa^{\prime}}\times 1, B)$ of the product space $X\times Y$ is constructed. Here, the indexing set $B$ is obtained by means of the set $A$ and certain subfamilies of $\Phi(a)=\{\varphi|\varphi:\mathscr{U}_{a}\rightarrow Cov(Y)\}, a\in A$, while the mesh $\mathscr{U}_{a} \times \varphi[\mathscr{U}_{a}]$ is a stacked covering of $X_{a}\times Y$ over $\mathscr{U}_{a}$.