Insertion in spaces, bispaces, ordered spaces and point-free spaces
by
Jorge Picado
University of Coimbra, Portugal
Coauthors: Maria João Ferreira and Javier Gutiérrez García

Insertion theorems on the existence of continuous real functions separating lower and upper semicontinuous real functions (like the Katetov-Tong insertion theorem characterizing normal spaces and the Stone insertion theorem characterizing extremally disconnected spaces) rank among the fundamental results in point-set topology. In this talk we present results of that kind for biframes (the point-free counterpart of bitopological spaces) that generalize at once the classical result and its counterparts in bitopological spaces (Priestley [6]), ordered topological spaces (Nachbin [4], Priestley [6]) and frames (recently obtained in [5], [3] and [1]; see also [2]).

References

[1] J. Gutiérrez García, T. Kubiak and J. Picado, Lower and upper regularizations of frame semicontinuous real functions, Algebra Universalis, to appear.

[2] J. Gutiérrez García, T. Kubiak, and J. Picado, Localic real-valued functions: a general setting, Preprint 08-20 of DMUC, April 2008 (submitted for publication).

[3] J. Gutiérrez García and J. Picado, On the algebraic representation of semicontinuity, J. Pure Appl. Algebra 210 (2007) 299-306.

[4] L. Nachbin, Topology and Order, Van Nostrand Mathematical Studies 4, Princeton, 1965.

[5] J. Picado, A new look at localic interpolation theorems, Topology Appl. 153 (2006), 3203-3218.

[6] H. Priestley, Separation theorems for semi-continuous functions on normally ordered topological spaces, J. London Math. Soc. (2) 3 (1971) 371-377.

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Date received: June 2, 2008