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. (shrink)

The present paper aims at examining the characteristics that determine the essential nature of the homogeneous bodies in Aristotle, from an analysis of Meteorology IV.12, which would at the same time establish a certain relationship with other treatises of natural philosophy and also in particular with books VII and VIII of Metaphysics. With this investigation, I will delineate a certain line of argument that goes against a reading perspective considered as traditional, with certain interpretive variants, according to which Aristotle would (...) have adopted the idea of ​​a universal teleology, in the general sense that all natural bodies would be generated for a specific goal, or for a natural purpose. According to a certain view, linked to this perspective, the teleological character of functional type of vital activities, notably expressed by the compositional arrangement of the non-homogeneous parts in the living being's complexion, would be somehow involved in the constitution of the homogeneous bodies considered in themselves and for themselves. In contrast to such a view, I will examine, in a precise way, to what extent the homogeneous bodies would comprise a certain formal factor directly involved in the characterization of their constitutional particularities, taking into account a comparative examination with other kinds of natural compositions, namely, elemental aggregates and living organisms. Thus, through this examination, I will explore the question of whether, in function of this formal factor, such bodies could present some teleological character trait, distinct from the functional type, characteristic of organic-animate constitutions. (shrink)