The 45th Spring Topology and Dynamical Systems Conference, 2011
We offer a mathematical logic framework for talking about when a given topological space is characterizable relative to a “class of its peers.” The framework involves a relational alphabet for which there is a natural way of assigning relational structures to spaces in the peer class; and a putative characterization of the given space consists of a set of first-order sentences over that alphabet, all true for the space. The attempted characterization is a success if any peer space satisfying all the sentences is inevitably homeomorphic to the given space.
For example, it has long been known that the arc may be characterized relative to the peer class of metrizable spaces using a single first-order sentence that holds for its bounded lattice of closed sets. Here we show that the arc may be characterized relative to the same class using the weaker language of betweenness. This involves a ternary relation on the set of points of the space: a lies between b and c just in case every closed connected subset containing both b and c must also contain a. We conjecture that trees can be so characterized, but not much else.
Bankston, Paul, "Characterizing the Arc" (2011). Mathematics, Statistics and Computer Science Faculty Research and Publications. 17.