In a previous paper [Luo 1995a] we presented a general theory of metric sites emphasizing its geometric applications. A metric site defined in that paper is a pair (C, t) of a category C and a metric functor t from C to the category Top of topological spaces. Replacing Top by the category Loc of locales, we obtain the notion of a framed site. In º0 - º2 we present an intrinsic approach to the theory of framed sites using sieves. Many notions and definitions introduced in [Luo 1995a] for metric sites can be extended to framed sites in a straightforward fashion. A key theorem concerning the Kan extension of a framed topology G is proved in º3, which is then applied to prove that any strict framed site has a completion in º4. 

Metric sites and Grothendieck toposes provide two important classes of framed sites. A metric site (C, t) determines a spatial framed site (C, O) where O: C ® Loc is the composition of t with the canonical functor Top ® Loc. Since Top is also a framed site, formally we can define a metric site to be an abstract framed site C together with a bicontinuous functor t from C to the framed site Top. An advantage of switching to (C, O) is that the theory of Kan extension for O is very simple. On the other hand the subobject classifier of a topos E with colimits determines a functor v_E: E ® Loc such that (E, v_E) is a complete canonical framed site (Theorem 9.3). 
The language of framed sites developed here can be applied to study the geometric properties of a topos. 

Another type of framed sites comes from Loc itself. Any category C with colimits determines a complete framed site Loc(C), consisting of the locales (A, O_A) over C, where A is a locale and O_A is a strict functor from A to C. It is a geometric closure of C in the sense that any strict framed site (C, G) can be embedded in Loc(C). Thus Loc(C) contains a completion of (C, G). This geometric approach to the theory of complete framed sites is covered in º5 - º8.