Chapter 13 - On D-spaces

Abstract

This chapter provides an overview of D-spaces. The chapter explains that a topological space X is a D-space if for every neighborhood assignment {N(x) : x ∈X} (that is, N(x) is an open neighborhood of x for each x ∈X) there is a closed discrete subset D of X such that X =U {N(x) : x ∈D}. A space X is sub-paracompact if every open covering of X can be refined by a σ-discrete closed covering. This chapter elaborates that the current state of knowledge about D-spaces is full of asymmetries. There are many theorems that state that certain types of spaces are D-spaces, but there are only a few theorems of the form “If X is a D-space, then . ” There are many results that state that spaces with certain types of bases are D-spaces, but there are no substantial theorems saying that spaces satisfying certain covering properties are D-spaces. There are fairly, general techniques for proving that something is a D-space, but more techniques are needed for building spaces that are not D-spaces. Correcting these asymmetries should provide the next generation of general topologists with ample work.