  We prove that for all countable tournaments $D$ the recently discovered
compactification $|D|$ by their ends and limit edges contains a topological
Hamilton path: a topological arc that contains every vertex. If $D$ is strongly
connected, then $|D|$ contains a topological Hamilton circle. These results
extend well-known theorems about finite tournaments, which we show do not
extend to the infinite in a purely combinatorial setting.
