What is a purely topological characterisation of the real line( standard topology)?



Topological Characterisation of the real line.
Here are a few examples that came up in a first search. Ward in "The topological characterization of an open linear interval", Proc. London Math. Soc.(2) 41 (1936), 191-198 proved the characterization of the real line as a connected, locally connected separable metric space, such that every point is a strong cut point (removing it leaves precisely two connected components). Franklin and Krishnarao proved that in this characterization "metric space" can be relaxed to "regular space", "On the topological characterization of the real line", Department of Mathematics, Carnegie-Mellon University, Report #69-36, 1969.


On a different note, Thron and Zimmerman prove in "A characterization of order topologies by means of minimal T0-topologies", Proc. Amer. Math. Soc. 27, (1971), 161-167, that order topologies $\tau$ on a set $X$ can be characterized as the topologies for which $(X,\tau)$ is $T\_1$ and $\tau$ is the least upper bound of two minimal $T\_0$ topologies. (Minimal here means that the open sets form a nested family of sets and that the complements of the point closures form a base for the topology.) Similarly the reals can be characterized as a connected, separable, $T\_1$ space, and $\tau$ is the least upper bound of two noncompact minimal $T\_0$ topologies.


