In their paper A Method for Constructing Ordered Continua , Hart and van Mill give the following definition of ordered continuum:

An ordered continuum is a compact, connected linearly ordered topological space, equivalently, a complete and densely linearly ordered set equipped with the order topology.

My question is a about the word complete.

I know that a linearly ordered space is connected if and only if it is densely ordered and Dedekind complete (every non empty subset that is bounded above admits a supremum). But that is not enough for this definition.

So I think that complete here means complete lattice (every subset has both supremum and infimum), as complete lattices are compact. Is that correct?

If it is, I have another question. Is completeness as a lattice equivalent to completeness in the sense of uniformity (in case the space is indeed uniformizable)?

Thank you!


Yes, "complete" here means "complete as a lattice". Alternatively, it just means the variant of Dedekind-complete where you require that any set has a supremum (without requiring the set to be nonempty or bounded).

Completeness in this sense is not equivalent to completeness with respect to a uniformity. For instance, $\mathbb{R}$ is complete with its standard uniformity, but is not complete as an ordered set in this sense.


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