Page 12 - Textos de Matemática Vol. 41

x Preface
Furthermore again, it has turned out that under very weak conditions (e.g., sobriety, or a separation property between T0 and T1 [2]) a space is already determined by the isomorphism type of its lattice of open sets (cf. [16], [64], articles that should have had more attention in their time).
Thus, the question naturally arises whether one cannot, after all, study a space as a complete lattice (with a special property — this had to be, of course, properly contemplated; it turned out that a certain distributive law does excellently, see the notion of a frame in 1.5 below).
The next question, however, is how to represent continuous maps. Lattice homomorphisms are not suitable, and neither are the complete lattice homo- morphisms; it turned out that one can represent the continuous maps by a sort of semi-complete lattice homomorphisms (frame homomorphisms, preserving all suprema and finite infima, see 2.1 below).
Observing that this is the right notion was an important breakthrough in the theory that started in late fifties and early sixties (let us mention the pio- neering articles [22], [47], [19]) and sent it on the track it has followed until today. (It should be noted that there were serious efforts, but not quite successful, to do topology without points much earlier, see, e.g., [43], [44], [66].)
Thus we have the following situation. Spaces are replaced by the above mentioned special complete lattices (frames); in particular, a classical space X is represented by its lattice Ω(X) of open sets. Some of the frames are not obtained from classical spaces, but the fact that we have “more spaces than before” is useful (although this was not obvious at the beginnings). The role of continuous maps is taken over by frame homomorphisms and everything is easy to work with. Moreover, for the important class of sober spaces (containing, e.g., all Hausdorff spaces, and all the finite ones) this representation is full and faithful, that is, frame homomorphisms between frames of open maps are in a natural one-to-one correspondence with the continuous maps between the spaces in question.
There is, however, an irritating drawback in this representation. Namely, it is contravariant, that is, this one-to-one correspondence associates continuous maps X → Y with frame homomorphisms Ω(Y ) → Ω(Y ). To mend it one usually turns the morphisms over formally, and considers, as the category of generalized spaces, the category Frmop dual to Frm (usually referred to as the category of locales). Thus we have localic morphisms L → M that are in fact frame homomorphisms L ← M. This helps a bit, but the intuition is, let us venture to say, not quite optimal. Instead of this elegant solution one might prefer a more down-to-earth one: a representation by suitable maps L → M . And this can be done. Frame homomorphisms h : M → L, by virtue of preserving all suprema, have uniquely defined right Galois adjoints f : L → M, meet- -preserving maps satisfying an extra condition (their left adjoints f∗ have to preserve finite meets).