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 iff X is countably metacompact. In a recent paper, Leiderman and Szeptycki studied Δ -spaces, which is a superclass 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 κ for which XL is not countably metacompact, in particular, not a Δ -space. We prove that an affirmative answer holds for the cardinal κ = cf( ω +1) . Assuming _ω = _ω , we get an example at a much lower cardinal, namely κ =2^2^2₀ , and our ladder system L is moreover ω -bounded.