The category of locales, which is the object of study of pointfree topology, has an algebraic dual, the category of frames. This fact was used by Joyal in order to introduce the frame of reals as a pointfree counterpart of the real line . Further, this lead to a fruitful study of the pointfree version of continuous real functions . Later several variants were introduced taking advantage of the presentation by generators and relations, for instance, the frame of extended reals , the frames of upper and lower reals, and the frame of partial reals introduced in  which arose naturally in the construction of the Dedekind completion of the lattice of continuous real functions on a frame.
A natural question arises: how does the topology of the unit circle and its group structure fit in all those variants? The aim of this talk is to try to address this question by giving a presentation of the frame of the unit circle and showing how its localic group structure is obtainable from the algebraic operations of the frame of reals.
Joint work with Javier Gutiérrez García and Jorge Picado.
 B. Banaschewski, The real numbers in pointfree topology, Textos de Matemática vol. 12, Departamento de Matemática da Universidade de Coimbra (1997).
 B. Banaschewski, J. Gutiérrez García, J. Picado, Extended real functions in pointfree topology, J. Pure Appl. Algebra 216 (2012) 905-922.
 A. Joyal, Nouveaux fondaments de l’analyse. Lectures Montréal 1973 and 1974 (unpublished).
 I. Mozo Carollo, J. Gutiérrez García, J. Picado, On the Dedekind completion of function rings, Forum Math. (doi: 10.1515/forum-2012-0095, in press).