Locally ordered topological spaces

Piotr Pikul

Abstrakt

While topology given by a linear order has been extensively studied, this cannot be said about the case when the order is given only locally. The aim of this paper is to fill this gap. We consider relation between local orderability and separation axioms and give characterisation of those regularly locally ordered spaces which are connected, locally connected or Lindel¨of. We prove that local orderability is hereditary on open, connected or compact subsets. A collection of interesting examples is also offered.

AMS Subject Classification: primary 54F05; secondary 54E99, 54D10, 06F30

Słowa kluczowe: order topology, linear order, separation axioms, connected space, local connectedness, compact space, Lindel¨of space, hereditary property
References

[1] H. Bennett, D. Lutzer, Recent Developments in the Topology of Ordered Spaces. In: Recent progress in general topology, II, North-Holland, Amsterdam, 2002, pp. 83–114.

[2] R. Engelking, General Topology. Revised and completed edition. Heldermann Verlag, Berlin 1989.

[3] H. Herrlich, Ordnungsf¨ahigkeit topologischer R¨aume. Dissertation, Berlin 1962.

[4] M.S. Kurili´c, A. Pavlovi´c, Topologies given by closed intervals. Novi Sad J. Math. 35:1 (2005), 187–195.

[5] D.J. Lutzer, A metrization theorem for linearly ordered spaces. Proc. Amer. Math. Soc. 22 (1969), 557–558.

[6] L. Nachbin, Topology and order. Translated from the Portuguese by Lulu Bechtolsheim. Van Nostrand Mathematical Studies, No. 4 D. Van Nostrand Co., Inc., Princeton, N.J.–Toronto, Ont.–London 1965

[7] S. Purisch, A History of Results on Orderability and Suborderability. In: Handbook of the History of General Topology. History of Topology, vol 2. (Eds. Aull C.E., Lowen R.) Springer, Dordrecht 1998, pp. 689–702.

[8] Yu.M. Smirnov, On metrization of topological spaces. Uspehi Mat. Nauk 6 (1951), 100–111.

[9] J.A. Steebach Jr., L.A. Steen, Counterexamples in Topology. 2nd edition. Springer- Verlag, New York, Heidelberg, Berlin 1978.

[10] L.A. Steen, A direct proof that a linearly ordered space is hereditarily collectionwise normal. Proc. Amer. Math. Soc. 24 (1970), 727–728.

[11] S. Willard, General Topology., Addison-Wesley Publishing Company 1970.

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