I was thinking about a counter example of a set that isn't a linear continuum because it doesn't satisfy the least upper bound property with the dictionary topology, but I can't found anyone.
2026-04-02 07:09:07.1775113747
Counter example of linear continuum.
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As I noted recently, $[0,1] \times [0,1)$ is not a linear continuum as $\{0\} \times [0,1)$ has an upper bound but no lub.