# 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.