Monotone insertion of continuous functions

Abstract

We consider the problem of inserting continuous functions between pairs of semicontinuous functions in a monotone fashion. We answer a question of Pan and in the process provide a new characterization of stratifiability. We also provide new proofs of monotone insertion results by Nyikos and Pan, and Kubiak. We then investigate insertion theorems for hedgehog-valued functions providing monotone versions of two theorems due to Blair and Swardson. From this we provide new characterizations involving hedgehogs of monotonically normal spaces, stratifiable spaces, normal, countably paracompact spaces, and perfectly normal spaces. The proofs are mostly geometric in nature.