Topological and Uniform Spaces pp 140-147 | Cite as

# Functional Separation Conditions

Chapter

## Abstract

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].

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## Copyright information

© Springer-Verlag New York Inc. 1987