Page 11 - Textos de Matemática Vol. 41

Preface
There are various reasons and motivations for the point-free approach to space. We are not historians and cannot (and do not really wish to) give in this short introduction a summary of the colorful development, where the impulses from logic, special results of set-theoretical topology, partially ordered sets, algebra, and other branches of mathematics united into the point-free topology as we know it today. For an excellent overview of the early stages we can warmly recommend the Johnstone’s article “The point of pointless topology” [36].
Here we want to emphasize the “realistic geometric” aspect in which one views a space as a structure of places of non-trivial extent rather than a set of points endowed with that or other structure. A result of a measurement is never a real number (one has a value ± a tolerance, which may be viewed as an open interval, hopefully very small, around the ideal value), or, pinpointing a position in a space (we now use the word “space” loosely) is pointing out a spot that is by no means infinitesimally small (although small enough to serve the task), not a “point” as we know it from classical geometry or topology. We can think of a point as an ideal limit of diminishing spots, and it is useful; but from our point of view it is secondary, and often not necessary (and not even advisable).
There are two big moments that initiated the development of modern theory of space that we call topology. One of them is very well defined: the Hausdorff’s “Mengenlehre” [27] of 1914 where the idea of space is based on the concept of neighbourhood, that then opens a general perspective for concepts like closure, interior, open and closed sets. The other, much less defined in time (one can speak of a good part of the twenties), is the tendency to give more and more importance (and priority when defining the other concepts) to open sets. Gradually one started to think of a topological space as of a set endowed with a system of subsets called open sets, with very simple properties, from which one could easily derive the other concepts one might wish to have.
Now the concept of an open set fits very well to the (rather unprecise) idea of a “spot in space” mentioned above. Furthermore, the structure of open sets is simple, a complete lattice (with a special property that is easy to understand and enhances the computational potential).
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