Can the reals be characterized by topological properties?

They are the unique complete ordered field. Completeness is a topological property (complete in the euclidean metric).

Any ordered field can be considered connected if one simply uses the trivial topology.

 

Given the standard Euclidean metric, then yes, properties such as the "least upper bound property", "monotone convergence", and the "Cauchy Criterion", all equivalent to "completeness", can be proven from the fact that the Real numbers are connected. As matt grime said, any set is connected in the trivial topology (where the only open sets are the empty set and the entire set).

Actually, both connectedness and the Heine-Borel theorem (that all closed and bounded sets are compact) can be shown to be equivalent to mono-tone convergence, least upper bound property, and Cauchy criterion. That is, given any one, you can prove the others. (Again, assuming the Eucliden metric: d(x,y)= |x- y|.)