Insertion of continuous set-valued mappings

Abstract

An interesting result about the existence of “intermediate” set-valued mappings between pairs of such mappings was obtained by Nepomnyashchii. His construction was for a paracompact domain, and he remarked that his result is similar to Dowker's insertion theorem and may represent a generalisation of this theorem. In the present paper, we characterise the τ-paracompact normal spaces by this set-valued “insertion” property and for τ=ω, i.e. for countably paracompact normal spaces, we show that it is indeed equivalent to the mentioned Dowker's theorem. Moreover, we obtain a similar result for τ-collectionwise normal spaces and show that for normal spaces, i.e. for ω-collectionwise normal spaces, our result is equivalent to the Katětov-Tong insertion theorem. Several related results are obtained as well.