On a problem of P(\(\alpha\) ,\(\delta\) ,\({\mathfrak n})\) concerning generalized Alexandroff's cube (Q1112361)

Universality of generalized Alexandroff's cube \(B^{{\mathfrak n}}_{\alpha,\delta}\) plays an essential role in the theory of absolute retracts for the category of \(<\alpha,\delta >\)-closure spaces. Alexandroff's cube \(B^{{\mathfrak n}}_{\alpha,\delta}\) is an \(<\alpha,\delta >\)-closure space generated by the family of all complete filters in a lattice of all subsets of a set of power \({\mathfrak n}\). Condition P(\(\alpha\),\(\delta\),\({\mathfrak n})\) says that \(B^{{\mathfrak n}}_{\alpha,\delta}\) is a closure space of all \(<\alpha,\delta >\)- filters in the lattice \(<{\mathcal P}({\mathfrak n}),\subseteq >\). Assuming that P(\(\alpha\),\(\delta\),\({\mathfrak n})\) holds, in the paper by \textit{A. W. Jankowski} [Stud. Logica 45, 155-166 (1986; Zbl 0612.54017)] there are given sufficient conditions saying when an \(<\alpha,\delta >\)-closure space is an absolute retract for the category of \(<\alpha,\delta >\)- closure spaces (see Theorems 2.1 and 3.4 in Jankowski's paper). It seems that, under assumption that P(\(\alpha\),\(\delta\),\({\mathfrak n})\) holds, it will be possible to give a uniform characterization of absolute retracts for the category of \(<\alpha,\delta >\)-closure spaces. Except Lemma 3.1 of \textit{A. W. Jankowski} [Stud. Logica 45, 135-154 (1986; Zbl 0622.54012)], there is no information when the condition P(\(\alpha\),\(\delta\),\({\mathfrak n})\) holds or when it does not hold. The main result of this paper says, that there are examples of cardinal numbers \(\alpha\), \(\delta\), \({\mathfrak n}\) such that P(\(\alpha\),\(\delta\),\({\mathfrak n})\) is not satisfied. Namely it is proved, using elementary properties of Lebesgue measure on the real line, that the condition \(P(\omega,\omega_ 1,2^{\omega})\) is not satisfied. Moreover it is shown that fulfillment of the condition is essential assumption in Theorems 2.1 and 3.4 of the second cited paper i.e. it cannot be eliminated.