Locale ($X,\mathcal{O}(X)$)

A locale is an object of the opposite of the category of frames. A map $f:X\to Y$ is specified by a frame homomorphism $f^{\ast }:\mathcal{O}(Y)\to \mathcal{O}(X)$. This makes open-set algebra primary and allows spaces without enough ordinary points. A point of a frame is the inverse-image map of a point of a locale.