Title: On Properties of Nests: Some Answers and Questions,
Abstract: 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.