On resolutions of linearly ordered spaces

Agata Caserta, Alfio Giarlotta, Stephen Watson
2006 Applied General Topology  
We define an extended notion of resolution of topological spaces, where the resolving maps are partial instead of total. To show the usefulness of this notion, we give some examples and list several properties of resolutions by partial maps. In particular, we focus our attention on order resolutions of linearly ordered sets. Let X be a set endowed 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 LOTS and ı : X ֒→ Y is
more » ... and ı : X ֒→ Y is an injective, order-preserving and open-in-the-range function. We exhibit a canonical unification (Y, ı) of (X, , τ ) such that Y 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 of X. Further, we explicitly describe the canonical unification of an order resolution. 2000 AMS Classification: 54F05, 06F30, 46A40, 54A10
doi:10.4995/agt.2006.1925 fatcat:k2c4uugapvhxhckdo4mqbsuzhe