Regularity and Fitness

Among the conditions of separation type, the axiom of regularity is very special. From the earliest stages of point-free topology there was natural interest in conditions that would capture classical separation phenomena as convincingly as possible in the new, more general, context. Classical separation axioms are typically formulated in the language of points and point-dependent notions; hence, one looked for equivalent formulations, or imitated the geometric intuition to obtain suitable replacements or at least analogies (see, e.g., Isbell (Math Scand 31:5–32, 1972), Dowker and Strauss. Separation axioms for frames. In: Topics in Topology, pp. 223–240. Proc. Colloq., Keszthely, 1972. Colloq. Math. Soc. Janos Bolyai, vol. 8, North-Holland, Amsterdam, 1974, Isbell (Math Scand 36:317–339, 1975), Simmons. A framework for topology. In: Logic Colloq. ’77, pp. 239–251. Stud. Logic Foundations Math., vol. 96. North-Holland, Amsterdam-New York, 1978, Johnstone. Stone Spaces. Cambridge Studies in Advanced Mathematics, vol. 3. Cambridge University Press, Cambridge 1982, Rosický and Šmarda (Math. Proc. Cambridge Philos. Soc. 98:81–86, 1985)). In this company, regularity stands out. As we have already seen in Chap. I, it can be translated very easily, and the obtained formula has a natural appeal even in classical spaces (in fact, in an obviously equivalent form it is used classically anyway). There is no reasonable doubt that this formula makes a fully satisfactory point-free extension. It can be used without problems for proving useful facts parallel with the classical ones; moreover, it is algebraically versatile and easy to work with.