3
$\begingroup$

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!

$\endgroup$
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.