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.