Spatial sublocales and essential primes (Q1090349)

It is well known that a locale is spatial precisely when every element is a meet of prime elements. Now if L is a spatial locale and a, p are elements of L with p prime and \(a\leq p\), p is called an essential prime of a when p is regular in the locale \(\uparrow a\). This is equivalent to the fact that p has to appear in any meet \(a=\wedge q\) where each q is a minimal element in the set of prime elements above a. For a locale L, the following conditions are proved to be equivalent: (1) Every sublocale of L is spatial, (2) every element of L is the meet of its essential prime elements, (3) the locale NL of nuclei of L is spatial. This result is particularized to the case of the locale of open subsets of the spectrum of a ring (i.e. the locale of radical ideals).