On resolutions of linearly ordered spaces

Agata Caserta, Alfio Giarlotta, Stephen Watson

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

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References

R. Engelking, General Topology (Heldermann Verlag, Berlin, 1989).

V. V. Fedorcuk, Bicompacta with noncoinciding dimensionalities, Soviet Math. Doklady, 9/5 (1968), 1148–1150.

K. P. Hart, J. Nagata and J.E. Vaughan (Eds.), Encyclopedia of General Topology (North-Holland, Amsterdam, 2004).

S. Watson, The Construction of Topological Spaces: Planks and Resolutions, in M. Husek and J. van Mill (eds.), Recent Progress in General Topology, 673–757 (North-Holland, Amsterdam, 1992).

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Universitat Politècnica de València

e-ISSN: 1989-4147   https://doi.org/10.4995/agt