Pointless topology

Definition

A frame is a complete lattice $L$ such that $a \wedge \bigvee_{i \in I} b_i = \bigvee_{i \in I} (a \wedge b_i)$ for every $a \in L$ and every family $(b_i)_{i \in I}$ in $L$. A frame homomorphism is a map between frames that preserves finite meets, including the top element, and arbitrary joins, including the bottom element. The category of locales is the opposite of the category of frames and frame homomorphisms. Pointless topology is the study of locales. Each topological space $X$ gives the locale $\Omega(X)$, the frame of open sets of $X$ ordered by inclusion. Each continuous map $f \colon X \to Y$ gives the frame homomorphism $f^{-1} \colon \Omega(Y) \to \Omega(X)$, hence a locale map $\Omega(X) \to \Omega(Y)$.