On resolutions of linearly ordered spaces


  • Agata Caserta Seconda Università degli Studi di Napoli
  • Alfio Giarlotta Università di Catania
  • Stephen Watson York University




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


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.


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Author Biographies

Agata Caserta, Seconda Università degli Studi di Napoli

Dipartimento di Matematica

Alfio Giarlotta, Università di Catania

Department of Economics and Quantitative Methods

Stephen Watson, York University

Department of Mathematics and Statistics


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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).


How to Cite

A. Caserta, A. Giarlotta, and S. Watson, “On resolutions of linearly ordered spaces”, Appl. Gen. Topol., vol. 7, no. 2, pp. 211–231, Oct. 2006.



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