A quotient-like construction involving elementary submodels

Abstract

This article is an investigation of a method of deriving a topology from a space and an elementary submodel containing it. We first define and give the basic properties of this construction, known as X/M . In the next section, we construct some examples and analyse the topological relationship between X and X/M . In the final section, we apply X/M to get novel results about Lindelöf spaces, giving partial answers to a question of F.D. Tall and another question of Tall and M. Scheepers.1 Definitions and preliminaries.We assume all spaces are T 3 1 2 -since we are primarily interested in strengthenings of the Lindelöf property, this causes no serious loss of generality. Given an elementary submodel M ≺ H λ and a space X with X ∈ M , we will refer to the classical notion of 'subspace with respect to M ' as X M . Recall that this is the topology on X ∩ M generated by the family (U ∩ M : U ∈ M is open in X) [7]. Our object of study is X/M , which can be seen as analogous to X M where 'subspace' is replaced by 'quotient'. The construction of X/M was introduced independently by I. Bandlow in [1] and A. Dow in [2], and discussed further by T. Eisworth in [3]. There are two related ways to define X/M ; we give the more elementary method first.Definition 1.1. Define an equivalence relation on X by letting x 0 ∼ x 1 if and only if f (x 0 ) = f (x 1 ) for all continuous f : X → R such that f ∈ M . Let X/M be the resulting quotient set and write π : X ։ X/M for the projection. We topologize X/M by taking a base to be all sets of the form π(U ), where U ∈ M is a cozero set in X.Definition 1.2. Since X is T 3 1 2 , we can consider the natural embedding e : X ֒→ [0, 1] C * (X) given by e(x) = (f (x)) f ∈C * (X) . There is also a natural mapThe equivalence of these formulations is proven in [3], with the correspondence given by sending the equivalence class [x] ∈ X/M of a point in x ∈ X to F • e(x). We will use them interchangably. We now summarize the relevant basic properties of X/M .