Generalization of Scott's formula for retractions from generalized Alexandroff's cube

Abstract

In the paper [2] the following theorem is shown: Theorem (Th. 3,5, [2]), If α=0 or δ=∞ or α⩾δ, then a closure space X is an absolute extensor for the category of 〈α, δ〉 -closure spaces iff a contraction of X is the closure space of all 〈α, δ〉-filters in an 〈α, δ〉-semidistributive lattice.

In the case when α=ω and δ=∞, this theorem becomes Scott's theorem:

Theorem ([7]). A topological space X is an absolute extensor for the category of all topological spaces iff a contraction of X is a topological space of “Scott's open sets” in a continuous lattice.

On the other hand, when α=0 and δ=ω, this theorem becomes Jankowski's theorem:

Theorem ([4]). A closure space X is an absolute extensor for the category of all closure spaces satisfying the compactness theorem iff a contraction of X is a closure space of all filters in a complete Heyting lattice.

But for separate cases of α and δ, the Theorem 3.5 from [2] is proved using essentialy different methods.

In this paper it is shown that this theorem can be proved using, for retraction, one uniform formula.

Namely it is proved that if α= 0 or δ= ∞ or α ⩾ δ and $F_{\alpha ,\delta } \left( L \right) \subseteq B_{\alpha ,\delta }^\mathfrak{n}$ and if L is an 〈α, δ〉-semidistributive lattice, then the function

$$r:{\text{ }}B_{\alpha ,\delta }^\mathfrak{n} \to F_{\alpha ,\delta } \left( L \right)$$

such that for x ε ℘ ($\mathfrak{n}$): (*)

$$r\left( x \right) = inf_L \left\{ {l \in L|\left( {\forall A \subseteq L} \right)x \in C\left( A \right) \Rightarrow l \in C\left( A \right)} \right\}$$

defines retraction, where C is a proper closure operator for $B_{\alpha ,\delta }^\mathfrak{n}$.

It is also proved that the formula (*) defines retraction for all 〈α, δ〉, whenever L is an 〈α, δ〉 -pseudodistributive lattice.

Moreover it is proved that when α=ω and δ=∞, the formula (*) defines identical retraction to the formula given in [7], and when α = 0 and δ=ω, the formula (*) defines identical retraction to the formula given in [4].