On resolutions of linearly ordered spaces

Abstract

We define an extended notion of resolution of topologicalspaces, where the resolving maps are partial instead of total. To showthe usefulness of this notion, we give some examples and list severalproperties of resolutions by partial maps. In particular, we focus ourattention on order resolutions of linearly ordered sets. Let X be a setendowed with a Hausdorff topology τ and a (not necessarily related)linear order . A unification of X is a pair (Y, ı), where Y is a LOTSand ı : X →֒ Y is an injective, order-preserving and open-in-the-rangefunction. We exhibit a canonical unification (Y, ı) of (X,, τ ) such thatY is an order resolution of a GO-space (X,, τ ∗), whose topology τ ∗refines τ . We prove that (Y, ı) is the unique minimum unification ofX. Further, we explicitly describe the canonical unification of an orderresolution.

Keywords

• Resolution
• Lexicographic ordering
• GO-space
• Linearly ordered topological space
• Pseudo-jump
• TO-embedding
• Unification