Page 15 - Textos de Matemática Vol. 41

Chapter 1 Preliminaries
In this chapter we will first recall and prove, when necessary, a few easy facts about posets, Heyting algebras, pseudocomplements and complements. All of them are indeed very simple but also very useful, as the reader will see when applying them in proofs along the text.
Further, we will introduce the notion of a frame (a complete lattice with a specific distributive law) that will prove to be crucial further on. We will then briefly discuss special filters and meet-irreducibility in this type of lattice.
Finally we will recall the notion of sobriety of a topological space, a stan- dard notion that is sometimes missing in standard textbooks.
1.1. Posets. Only basic facts about posets (i.e. partially ordered sets) are as- sumed; if necessary, the reader can consult, e.g., [18]. The notation is the usual one, for instance we write for a subset M or an element x of a poset (X, ≤),
↓M = {x|x≤m∈M}, ↓x=↓{x}, ↑M = {x|x≥m∈M}, ↑x=↑{x}.
1.2. (Galois) adjunctions. Monotone maps
f : (X,≤) → (Y,≤), g : (Y,≤) → (X,≤)
are (Galois) adjoint (f on the left and g on the right) if f(x)≤y iff x≤g(y)
(equivalently, if fg(y) ≤ y and x ≤ gf(x) for every x ∈ X and y ∈ Y ). Recall the well-known fact that
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