The Topological Characterisation of an Open Linear Interval

On Countable Dense and n-homogeneity

“…The following result is the first step in the proof of Ward's Theorem A in [12]. Its proof is repeated here for the sake of completeness.…”

“…If X \ {p} is connected for some p ∈ X, then we are done by Theorem 3.4. If not, then by Ward [12] we arrive at the conclusion that X is homeomorphic to R, and hence there is nothing to prove.…”

“…So we may assume that X \ {p} has exactly two components for some (equivalently: every) p ∈ X. Hence if X were locally connected, then by Ward [12] we would get that X is homeomorphic to the real line R and one could then try to use its properties to complete the proof. However, it is unknown whether a Polish connected CDH-space is locally connected (Fitzpatrick and Zhou [5,Problem 386]), hence this approach does not seem to work.…”

A Topological Variation of the Reconstruction Conjecture

“…Proof. The reals: Ward proved in [12] that every metric, locally connected, separable connected space, where every card has exactly two components, is homeomorphic to the space of real numbers. By Theorems 6.4, 3.3 and 4.2, any reconstruction of R is metrizable, locally connected and separable.…”

Euclidean representations of topologically ordered spaces

“…Thus, Theorem 12.3 is the analogue of a theorem of A. J. Ward [10], wherein he characterized R topologically, but in a manner closely related to the order-properties of (R, =s). We use the frontiers of the sets V x , A x to form sets which (with the right axioms) have the separation properties of coordinate planes.…”