Applied Categorical Structures (2005) 13: 281–303 © Springer 2005 DOI: 10.1007/s10485-005-0355-8 Functorial Quasi-Uniformities on Frames MARIA JOÃO FERREIRA and JORGE PICADO ⋆ Departamento de Matemática, Universidade de Coimbra, 3001-454 Coimbra, Portugal. e-mail: {mjrf,picado}@mat.uc.pt (Received: 25 June 2003; accepted: 28 June 2005) Abstract. We present a unified study of functorial quasi-uniformities on frames by means of Weil entourages and frame congruences. In particular, we use the pointfree version of the Fletcher con- struction, introduced by the authors in a previous paper, to describe all functorial transitive quasi- uniformities. Mathematics Subject Classifications (2000): 06D22, 54B30, 54E05, 54E15, 54E55. Key words: frame, quasi-uniform frame, Weil entourage, biframe, strictly zero-dimensional biframe, Skula biframe, Skula functor, functorial quasi-uniformity, T -pseudosection. 1. Introduction The method of constructing transitive compatible quasi-uniformities for an arbi- trary frame [11], extending classical results of Fletcher for quasi-uniform spaces [12], naturally raises the question of its functoriality. The purpose of the present paper is to address this question. To put this in perspective, we recall that a topological space (X, T ) is uni- formizable if there exists a uniformity E on X such that the corresponding induced topology T(E ) coincides with the given topology T . As is well-known, the topo- logical spaces that are uniformizable are precisely the completely regular ones. This result has a perfect analog in the two-sided theory of quasi-uniform spaces (where they are considered over their induced bitopologies): a bitopological space (X, T 1 , T 2 ) is quasi-uniformizable, i.e. there exists a quasi-uniformity E on X such that T(E ) = T 1 and T(E −1 ) = T 2 , if and only if it is pairwise completely regular. However, in the one-sided theory, where a quasi-uniformity is considered over a single underlying topology, the resemblance with the symmetric case is over and one gets a striking result: every topological space is quasi-uniformizable. Indeed, for every topological space (X, T ), there exists a (transitive) quasi-uniformity E P (T ) on X, admissible on (X, T ), that is, which induces as its first topology the given topology T ([9, 18]). The quasi-uniformity E P (T ) is nowadays called the Császár–Pervin quasi-uniformity. So, every topological space (X, T ) gives rise ⋆ The authors acknowledge partial financial assistance by the Centro de Matem´ atica da Universi- dade de Coimbra/FCT.