DIAMOND on LADDER SYSTEMS and COUNTABLY METACOMPACT TOPOLOGICAL SPACES

The property of countable metacompactness of a topological space gets its importance from Dowker's 1951 theorem that the product of a normal space X with the unit interval [0; 1] is again normal i δ X is countably metacompact. In a recent paper, Leiderman and Szeptycki studied δ-spaces, which are a subclass of the class of countably metacompact spaces. They proved that a single Cohen real introduces a ladder system L over the first uncountable cardinal for which the corresponding space XL is not a δ-space, and asked whether there is a ZFC example of a ladder system L over some cardinal δ for which XL is not countably metacompact, in particular, not a δ-space. We prove that an δ rmative answer holds for the cardinal ]ω+ = cf(]ω+1). Assuming i! =, we get an example at a much lower cardinal, namely κ = 2^22ℵ0, and our ladder system L is moreover !-bounded.