Completely normal frames and real-valued functions

• Author(s): Ferreira, Maria João ; Gutiérrez García, Javier ; Picado, Jorge
• Date: 2008
• Persistent ID: http://hdl.handle.net/10316/11233
• Origin: Estudo Geral - Universidade de Coimbra
• Subject(s): Frame; Localic real function; Normal frame; Completely normal frame; Completely normal space; Extremally disconnected frame; Insertion; Upper and lower regularizations
• Document Type: Preprint
• Language: English

Description

Up to now point-free insertion results have been obtained only for semicontinuous real functions. Notably, there is now available a setting for dealing with arbitrary, not necessarily (semi-)continuous, point-free real functions, due to Guti errez Garc a, Kubiak and Picado, that gives point-free topology the freedom to deal with general real functions only available before to point-set topology. As a rst example of the usefulness of that setting, we apply it to characterize completely normal frames in terms of an insertion result for general real functions. This characterization extends a well known classical result of T. Kubiak about completely normal spaces. In addition, characterizations of completely normal frames that extend results of H. Simmons for topological spaces are presented. In particular, it follows that complete normality is a lattice-invariant property of spaces, correcting an erroneous conclusion in [Y.-M. Wong, Lattice-invariant properties of topological spaces, Proc. Amer. Math. Soc. 26 (1970) 206-208]. Centre of Mathematics of the University of Coimbra/FCT; Ministry of Education and Science of Spain; FEDER MTM2006-14925-C02-02; University of the Basque Country under grant GIU07/27