Functional Separation Conditions
Let H, K be a disjoint pair of subsets of the topological space X. Chapter 6 is concerned with the question of whether H and K can be separated: Do there exist open neighbourhoods U, V of H, K, respectively, which are also disjoint? The present chapter is concerned with the question of whether H and K can be functionally separated: Does there exist a continuous real-valued function α: X →I such that α = 0 throughout H and α = 1 throughout K? The latter condition implies the former since we can take U = α-1[0,1-2, V = α-1(1-2, 1].
Unable to display preview. Download preview PDF.