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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 
$\Delta $-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 
$X_{ L}$ is not a 
$\Delta $-space, and asked whether there is a ZFC example of a ladder system 
$ L$ over some cardinal 
$\kappa $ for which 
$X_{ L}$ is not countably metacompact, in particular, not a 
$\Delta $-space. We prove that an affirmative answer holds for the cardinal 
$\kappa =\operatorname {\mathrm {cf}}(\beth _{\omega +1})$. Assuming 
$\beth _\omega =\aleph _\omega $, we get an example at a much lower cardinal, namely 
$\kappa =2^{2^{2^{\aleph _0}}}$, and our ladder system L is moreover 
$\omega $-bounded.
