×

Nests, and their role in the orderability problem. (English) Zbl 1383.54034

Rassias, Themistocles M. (ed.) et al., Mathematical analysis, approximation theory and their applications. Cham: Springer (ISBN 978-3-319-31279-8/hbk; 978-3-319-31281-1/ebook). Springer Optimization and Its Applications 111, 517-533 (2016).
Let \((X,\tau)\) be a topological space. Under what conditions does there exist an order relation \(\leq\) on \(X\) such that the open-interval topology \(\tau_{\leq}\) induced by \(\leq\) is equal to the given \(\tau\)?
This is the general orderability problem. Nests on \(X\), that is, collections \(\mathcal N\) of subsets of \(X\) such that for each \(A,B\in\mathcal{N}\), either \(A\subseteq B\) or \(B\subseteq A\), were used by J. van Dalen and E. Wattel [General Topology Appl. 3, 347–354 (1973; Zbl 0272.54026)] to solve the orderability problem in the case of \(T_1\)-spaces. The first part of the book chapter under review presents a survey on nests and the orderability problem. The second part discusses some new results, partial results and open questions.
For the entire collection see [Zbl 1348.00048].

MSC:

54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
54-02 Research exposition (monographs, survey articles) pertaining to general topology

Citations:

Zbl 0272.54026
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Brian, W.R.: Neight: the nested weight of a topological space. Topol. Proc. 47, 279–296 (2016) · Zbl 1329.54006
[2] de Groot, J., Schnare, P.S.: A topological characterization of products of compact totally ordered spaces. Gen. Topol. Appl. 2, 67–73 (1972) · Zbl 0254.54006 · doi:10.1016/0016-660X(72)90038-4
[3] Faber, M.J.: Metrizability in Generalized Ordered Spaces. Matematisch Centrum, Amsterdam (1974) · Zbl 0282.54017
[4] Giordano, P.: Fermat Reals: Nilpotent Infinitesimals and Infinite Dimensional Spaces. Prebook http://arxiv.org/abs/0907.1872
[5] Giordano, P.: The ring of fermat reals. Adv. Math. 225, 2050–2075 (2010) · Zbl 1205.26051 · doi:10.1016/j.aim.2010.04.010
[6] Giordano, P.: Fermat reals: infinitesimals without logic. Miskolc Math. Notes 14(3), 65–80 (2013) · Zbl 1299.14002
[7] Giordano, P., Kunzinger, M.: Topological and algebraic structures on the ring of fermat reals. Isr. J. Math. 193, 459–505 (2013) · Zbl 1275.26049 · doi:10.1007/s11856-012-0079-z
[8] Good, C., Papadopoulos, K.: A topological characterization of ordinals: van Dalen and Wattel revisited. Topol. Appl. 159, 1565–1572 (2012) · Zbl 1241.54021 · doi:10.1016/j.topol.2011.02.014
[9] Hart, K.P., Nagata, J.-I., Vaughan, J.E.: Encyclopedia of General Topology. Elsevier Science and Technology Books (2014) ISBN: 978-0-444-50355-8 · Zbl 1059.54001
[10] Papadopoulos, K: On properties of nests: some answers and questions. Quest. Answ. Gen. Topol. 33, 71–91 (2015) · Zbl 1333.54039
[11] Papadopoulos, K.: On the orderability problem and the interval topology. In: Rassias, T., Toth, L. (eds.) Topics in Mathematical Analysis and Applications. Optimization and Its Applications Springer Series. Springer, New York (2014)
[12] Purisch, S.: A History of Results on Orderability and Suborderability. Handbook of the History of General Topology, vol. 2 (San Antonio, TX, 1993), volume 2 of Hist. Topol., pp. 689–702. Kluwer Academic Publications, Dordrecht (1998) · Zbl 0933.54001 · doi:10.1007/978-94-017-1756-4_10
[13] Reed, G.M.: The intersection topology w.r.t. the real line and the countable ordinals. Trans. Am. Math. Soc. 247(2), 509–520 (1986) · Zbl 0602.54001
[14] van Dalen, J., Wattel, E.: A topological characterization of ordered spaces. Gen. Topol. Appl. 3, 347–354 (1973) · Zbl 0272.54026 · doi:10.1016/0016-660X(73)90022-6
[15] van Douwen, E.K.: On Mike’s misnamed intersection topologies. Topol. Appl. 51(2), 197–201 (1993) · Zbl 0827.54021 · doi:10.1016/0166-8641(93)90153-5
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.