Coherent locales

Let \(X\) be a locale. An element of \(X\) is compact if every expression of it as a join admits a finite subcover; equivalently, the corresponding subterminal object of \(\Shv(X)\) is quasi-compact. The topos \(\Shv(X)\) is locally coherent if and only if the compact elements form a basis which is closed under finite meets. In this case \(X\) is called a locally coherent locale. It is coherent precisely when, in addition, the top element is compact.

For locales, local coherence already implies local \(n\)-coherence for every \(n\). Consequently, the preceding conditions characterize locally coherent and coherent localic topoi, respectively. The completeness theorem below implies that a coherent locale is spatial, and its spatial realization is a spectral space. Conversely, the locale of open subsets of a spectral space is coherent. The corresponding local statement identifies locally coherent spatial locales with locally spectral spaces.