Frame

Remarks

Example: for a topological space $X$, the open sets of $X$ ordered by inclusion form a frame $\Omega(X)$; joins are unions, finite meets are intersections, and $U \cap \bigcup_i V_i = \bigcup_i (U \cap V_i)$. Non-example: the lattice of subspaces of $\R^2$ is complete but not a frame; for three distinct lines $a, b, c$ through $0$, $a \wedge (b \vee c) = a$ while $(a \wedge b) \vee (a \wedge c) = \set{0}$. A complete lattice is a frame if and only if it is a Heyting algebra, with implication $a \to b = \bigvee \set{c : a \wedge c \le b}$. Every frame is a distributive lattice. As ordered sets, frames, locales and complete Heyting algebras are the same objects; the three names differ in the maps one uses between them.