Locale

A complete Heyting algebra (see Brouwer lattice) regarded as a "generalized topological space" . The name "locale" is due to J.R. Isbell [a1], although the concept had been studied by a number of earlier writers: the basic idea is that, for any topological space $X$, the lattice $\mathcal{O}(X)$ of open subsets of $X$ is complete and satisfies the infinite distributive law $$ U \cap \bigcup_{i \in I} V_i = \bigcup_{i \in I} U \cap V_i $$ (equivalently, it is a Heyting algebra), and many important topological properties of spaces (compactness, connectedness, etc.) are in fact properties of their open-set lattices. Thus one may regard any complete lattice satisfying the infinite distributive law (such a lattice is commonly called a frame) as if it were the open-set lattice of a space, irrespective of whether it possesses enough "points" to be representable as an actual lattice of open subsets. A frame homomorphism is a mapping preserving finite meets (intersections) and arbitrary joins (unions). A locale is extensionally the same thing as a frame, but intensionally different: the difference resides in the fact that a morphism (or continuous mapping) of locales from $X$ to $Y$ is defined to be a frame homomorphism from $Y$ to $X$. (To emphasize the intensional difference, some authors write $\mathcal{O}(X)$ for the frame corresponding to a locale $X$. Other authors — e.g. those of [a2] — use a different terminology: they redefine "space" to mean what is called a locale above, and use "locale" to mean a frame in the terminology above. The sense in which "locale" is used in this article is the original one, as used by Isbell.)

A frame is representable as the open-set lattice of a space if and only if every element is expressible as a meet of prime elements; locales with this property are called spatial. The space corresponding to a spatial locale is not uniquely determined, but it becomes so if one requires that it be sober, i.e. that every prime open set should be the complement of the closure of a unique point. (Every Hausdorff space is sober, and every sober space satisfies the $T_0$ separation axiom.) The passage from sober spaces to locales is a full imbedding of categories; it does not preserve products in general, but it does so if one of the factors is locally compact (for a more general form of this result, see [a3]). Many familiar topological properties can be extended from the category of (sober) spaces to the category of locales. For example, one defines a locale to be locally compact if the corresponding frame is a continuous lattice; it can be shown [a4] that every locally compact locale is spatial, and in fact the category of locally compact locales is equivalent to that of locally compact sober spaces.

There is a notion of sublocale (or quotient frame) which corresponds to that of a topological subspace. Every locale can be represented as a sublocale of a spatial locale, but in general sublocales behave rather differently from subspaces: this is most evident in the fact that the intersection of any number of dense sublocales of a given locale is dense. It is this property (often in conjunction with the discrepancy between the two notions of product) which gives rise to most of the significant differences between the category of locales and that of spaces; as was first pointed out by Isbell [a1], these differences often have the effect of making the former a pleasanter category to work in than the latter. Three examples of this: the property of paracompactness for locales is inherited by arbitrary locale products [a1], the Lindelöf covering property for locales is equivalent to realcompactness [a5], and every localic subgroup of a localic group is closed [a6], [a7] (localic groups are to topological groups as locales are to spaces).

As well as localic groups, uniform locales have received some attention [a1], [a8]: a uniformity on a locale is usually defined as a family of coverings (i.e. subsets of the frame whose join is the top element) satisfying appropriate closure properties. Locales also play an important part in the constructive approach to general topology [a9], [a10]. For a general account of locale theory, see [a11].

References

[a1]	J.R. Isbell, "Atomless parts of spaces" Math. Scand. , 31 (1972) pp. 5–32 Zbl 0246.54028
[a2]	A. Joyal, M. Tierney, "An extension of the Galois theory of Grothendieck" Mem. Amer. Math. Soc. , 309 (1984)
[a3]	J.R. Isbell, "Product spaces in locales" Proc. Amer. Math. Soc. , 81 (1981) pp. 116–118
[a4]	B. Banaschewski, "The duality of distributive continuous lattices" Canad. J. Math. , 32 (1980) pp. 385–394
[a5]	J. Madden, J. Vermeer, "Lindelöf locales and realcompactness" Math. Proc. Cambridge Philos. Soc. , 99 (1986) pp. 473–480
[a6]	J.R. Isbell, I. Kříž, A. Pultr, J. Rosický, "Remarks on localic groups" F. Borceux (ed.) , Categorical Algebra and its Applications , Lect. notes in math. , 1348 , Springer (1988) pp. 154–172
[a7]	P.T. Johnstone, "A simple proof that localic subgroups are closed" Cahiers Top. et Géom. Diff. Catégoriques , 29 (1988) pp. 157–161
[a8]	A. Pultr, "Pointless uniformities" Comment. Math. Univ. Carolinae , 25 (1984) pp. 91–120
[a9]	M.P. Fourman, R.J. Grayson, "Formal spaces" A.S. Troelstra (ed.) D. van Dalen (ed.) , The L.E.J. Brouwer Centenary Symposium , Studies in logic , 110 , North-Holland (1982) pp. 107–122
[a10]	P.T. Johnstone, "Open locales and exponentiation" J.W. Gray (ed.) , Mathematical Applications of Category Theory , Contemp. Math. , 30 , Amer. Math. Soc. (1984) pp. 84–116
[a11]	P.T. Johnstone, "Stone spaces" , Cambridge Univ. Press (1982)