ABSTRACT
If X is an arbitrary T0-space and L the algebraic lattice Filt O(X) of all filters of open sets of X, then the function x ↦ u ( x ) : X → L https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072621/6e3dda93-0297-4e21-9483-9ad6f30dbbc7/content/eq1444.tif"/> L which associates with a point its neighborhood filter is a topological embedding if L is given its Scott topology σ(L). This was established by Banaschewski [1977] and used for the construction of the essential hull of X. Each neighborhood filter ty(x) is completely prime in L. We recall that an element p of a complete lattice L is called completely prime if and only if for any subset T of L the relation inf T ≤ p implies that t ≤ p for some t ∈ T. We shall denote the set of all completely prime elements of a complete lattice L by θ(l). It is an almost direct consequence of the definition that an element p of L is contained in θ(L) iff there is a compact element p. in L such that L is the disjoint union of ip and tp.. (The element p* is a completely coprime element, and the function p⊢p* is a bijection from Bel) onto the set of completely coprime elements.) Each p ∈ θ(L) determines a complete lattice morphism f:L → 2 via f − 1 ( O ) = ↓ p https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072621/6e3dda93-0297-4e21-9483-9ad6f30dbbc7/content/eq1445.tif"/> , and p ↦ f https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072621/6e3dda93-0297-4e21-9483-9ad6f30dbbc7/content/eq1446.tif"/> is a bijection from θ(L) onto (INF ∩ SUP)(L,2) in the terminology of A Compendium of Continuous Lattices [1980, p. 171, Definition 1.1].
