# Counter example of linear continuum.

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.

• $\mathbb Q$?  Dec 12 '19 at 15:01
• Well yes, but I'm searching for an counter example in $\mathbb{R^2}$
– user732763
Dec 12 '19 at 15:04
• $\mathbb Q \times \mathbb Q$? Dec 12 '19 at 15:13

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.