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.
Let , where is a measurable space, and a topological space. We study inclusions between three classes of extended real-valued functions on which are upper semicontinuous in and satisfy some measurability conditions.
In this note we are going to study dense covers in the category of locales. We shall show that any product of finitely regular locales with some dense covering property has this property as well.