NEW PROOFS OF CLASSICAL INSERTION THEOREMS

Abstract. We provide new proofs for the classical insertion theorems of Dowker and Michael. The proofs are geometric in nature and highlight the connection with the preservation of normality in products. Both proofs follow directly from the Katětov-Tong insertion theorem and we also discuss a proof of this. A function from a topological space X to R is said to be upper semi-continuous if, for every a in R, the preimage of [a,∞) is closed and lower semicontinuous if the preimage of (−∞, a] is closed. Given a pair of semicon-tinuous functions g ≤ h one can ask whether there is a continuous function f, with g ≤ f ≤ h. Such insertion results form part of the classical theory of general topology, tracing back to Hahn [5], who proved Theorem 1 in the realm of metrizable spaces, and Dieudonne ́ [2], who proved Theorems 1 and 2 for paracompact spaces. Theorem 1 (Katětov [7], Tong [16]). A space X is normal if and only if whenever g, h: X → R are upper (resp. lower) semi-continuous and g ≤ h, there is a continuous f: X → R such that g ≤ f ≤ h. Theorem 2 (Dowker [3]). A space X is normal and countably paracompact if and only if whenever g, h: X → R are upper (resp. lower) semi-continuous and g < h, there is a continuous f: X → R such that g < f < h. Theorem 3 (Michael [13]). A space X is perfectly normal if and only if whenever g, h: X → R are upper (resp. lower) semi-continuous and g ≤ h, there is a continuous f: X → R such that g ≤ f ≤ h and g(x) < f(x) < h(x) whenever g(x) < h(x). There is an intimate connection between insertion theorems and the nor-mality of product spaces, an area that has been the focus of much attention [14]. Normality is significant as it is precisely the property allowing contin-uous, real-valued functions on closed subspaces to be extended to the whole space. However, it does not behave well on taking products: for example it is known that a normal space X need not be binormal (that is, X × [0, 1] need not be normal) [4]. Binormality was, for a long time, an hypothesis in Borsuk’s homotopy extension theorem (Rudin and Starbird eventually proved that normality suffices [14]). The connection between binormalit