On a problem of P(α, δ, π) concerning generalized Alexandroff s cube

Abstract

Universality of generalized Alexandroff's cube $B_{\alpha ,\delta }^\mathfrak{n}$ plays essential role in theory of absolute retracts for the category of ťα, δ〉-closure spaces. Alexandroff's cube. $B_{\alpha ,\delta }^\mathfrak{n}$ is an ťα, δ〉-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(α, δ, $\mathfrak{n}$) says that $B_{\alpha ,\delta }^\mathfrak{n}$ is a closure space of all 〈α, δ〉-filters in the lattice 〈π($\mathfrak{n}$), $\subseteq$〉.

Assuming that P (α, δ, $\mathfrak{n}$) holds, in the paper [2], there are given sufficient conditions saying when an 〈α, δ〉-closure space is an absolute retract for the category of 〈α, δ〉-closure spaces (see Theorems 2.1 and 3.4 in [2]).

It seems that, under assumption that P (α, δ, $\mathfrak{n}$) holds, it will be possible to givean uniform characterization of absolute retracts for the category of 〈α, δ 〉-closure-spaces.

Except Lemma 3.1 from [1], there is no information when the condition P (α, δ, $\mathfrak{n}$) holds or when it does not hold.

The main result of this paper says, that there are examples of cardinal numbers, α, δ, $\mathfrak{n}$ such that P (α, δ, $\mathfrak{n}$) is not satisfied.

Namely it is proved, using elementary properties of Lebesgue measure on the real line, that the condition P (ω, ω ₁, 2^ω) is not satisfied.

Moreover it is shown that fulfillment of the condition is essential assumption in, Theorems 2.1 and 3.4 from [1] i.e. it cannot be eliminated.