On spaces without non-trivial subcontinua and the dimension of their products

We introduce splintered and strongly splintered spaces. They are generalizations of both almost zero-dimensional spaces and weakly 1-dimensional spaces. We prove that there are n-dimensional strongly splintered spaces for every n, and that there is a 1-dimensional splintered space X such that dimXn=n for every n. This solves a problem in the literature. Finally, we correct a flaw in an argument of Tomaszewski in his product formula for the dimension of the product of a weakly n-dimensional and a weakly m-dimensional space.