Page 21 - Textos de Matemática Vol. 41

Chapter 2
Locales, frames and spaces
2.1. Spaces and frames. Recall the frames from 1.5. We have already observed that they coincide, in essence, with the complete Heyting algebras, and noted that, nevertheless, the extra term is justified. Speaking of frames we have in mind a topological motivation, extrapolating the properties of the lattice Ω(X) of open sets of a topological space (recall 1.7) that satisfies the formula (f-distr) but no simple stronger one (simply because the finite meets are intersections, distributing over unions, while the infinite meets are generally distinct from intersections and have less transparent behavior). Further, if f : X → Y is a continuous mapping think of the natural maps defined by
Ω(f) : Ω(Y ) → Ω(X), Ω(f)(U) = f−1[U],
and observe that (because of the standard properties of the preimage function) they preserve all joins (that is, unions) but only finite meets (which, unlike the infinite ones, are intersections). Extrapolating from this observation, define frame homomorphisms as maps preserving all joins (including the bottom 0) and all finite meets (including the top 1).
(Thus, in this context we do not require preserving general meets, not to speak about the Heyting opera- tion.)
We will see shortly (in 2.5 below) that they provide a good model of continuity. The resulting category will be denoted by
Frm.
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