Extremally disconnected resolutions of $T_0$-spaces

The essential cover and the absolute cover of a schematic space

“…[4,Lemma 4]). Every essentially dense morphism in GSp is skeletal , and every skeletal morphism is regular.…”

“…A continuous map f : X → Y is said to be skeletal [4] if the inverse image f −1 (V ) of a dense V ∈ O(Y ) is dense in X. The subcategory of GSp with the same objects and skeletal maps as morphisms will be denoted by SSp.…”

The Absolute of a Topological Space and its Application to Abelian l-groups

“…We write B(X) for the complete Boolean algebra (see [15], 3.1) of regular open sets in X. The following result shows that perfect maps f : X → Y establish a strong relationship between the topologies of X and Y: They are r. o.-minimal in the sense of [4,29].…”

“…S. Aleksandrov called p the absolute of X. Iliadis [19] generalized the absolute to Hausdorff spaces, at the expense that p is no longer continuous. His construction was modified by Mioduszewski and Rudolf [29], which enabled Błaszczyk [4] to define a continuous absolute for T 0 -spaces. Finally, Ulyanov [38] generalized the concept to arbitrary topological spaces.…”

“…Apart from being perfect, an absolute p has to be surjective and "irreducible", where the latter concept has found several interpretations or replacements [3,4,29,31]. Our approach is categorical: Since perfect maps are closed, it makes no difference if "surjective" is replaced by dense, i.e.…”