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)?