The duality between normality and extremal disconnectedness from the algebraic pointfree point of view
14/03/2014 Sextafeira, 14 de Março de 2014, 14h15m, Sala B201
Jorge Picado (CMUC, Universidade de Coimbra)
Instituto para a Investigação Interdisciplinar da Universidade de Lisboa
Several familiar pairs of characterizations of the concepts of normality and extremal disconnectedness in classical topology show a remarkable "duality" between the two concepts. Nevertheless the proofs of the results in each pair are quite different in nature, requiring even in some cases different approaches and tools.
We will discuss the source of this duality in the more general setting of pointfree topology, taking advantage of its algebraic nature. We will show that each pair of parallel results can be framed by a single proof. The key ingredients will be relative notions of normality, extremal disconnectedness, semicontinuity and continuity (with respect to a fixed class of complemented sublocales) that bring into pointfree topology a variety of well known classical variants of normality and upper and lower semicontinuities in a nice unified manner.
Our results extend and unify the most relevant classical insertion, extension and separation results [Tietze (1915), Urysohn (1925), Stone (1949), Katetov (1951), Tong (1952), Gillman & Jerison (1960), Blatter & Seever (1972), Lane (1975,1979), etc.].
This talk is based on a paper with J. Gutierrez Garcia (Bilbao) that has just appeared online [On the parallel between normality and extremal disconnectedness, Journal of Pure and Applied Algebra 218 (2014) 784{803].
