2017/10/18 by Kyriakos Papadopoulos
Mathematics · #math.GN
published as Questions and Answers in General Topology, Volume 33, Number 2, 2015 · This paper has been published in the journal: Q & A in General Topology (Vol. 33, No 2, 2015.). arXiv admin note: substantial text overlap with arXiv:1302.2519
arxiv created 2017/10/18 · arxiv updated 2017/10/20
By considering nests on a given space, we explore order-theoretical and topological properties that are closely related to the structure of a nest. In particular, we see how subbases given by two dual nests can be an indicator of how close or far are the properties of the space from the structure of a linearly ordered space. Having in mind that the term interlocking nest is a key tool to a general solution of the orderability problem, we give a characterization of interlocking nest via closed sets in the Alexandroff topology and via lower sets, respectively. We also characterize bounded subsets of a given set in terms of nests and, finally, we explore the possibility of characterizing topological groups via properties of nests. All sections are followed by a number of open questions, which may give new directions to the orderability problem.