What is a link between the topological and order-theoretic completeness? Take $\mathbb{R}$ as an example.
Order-theoretically, the set $\mathbb{R}$ of all real numbers can be developed as a complete totally ordered field, where "complete" indicates that the supremum axiom is imposed. 
On the other hand, the set $\mathbb{R}$ happens to be topologically complete, in the sense that every Cauchy sequence in $\mathbb{R}$ converges in $\mathbb{R}$.
Is the order-theoretic completeness a notion developed independent of the concept of topological completeness, or is the converse true, or none of them is true? In general, what is a link between them?
 A: The two concepts are strictly linked through the modern definitions of real number.
We have to consider Cantor's definition, based on Cauchy sequence, and Dedekind's one, based on Dedekind cuts.
Several axioms has been proposed to ensure the so-called "completeness" like :


*

*metric completeness : every Cauchy sequence of points in a metric space $M$ has a limit in $M$

*order-theoretic completeness.
See also Construction of the real numbers, Dedekind's Contributions to the Foundations of Mathematics and The Early Development of Set Theory; both Cantor and Dedekind was at the origins of :


*

*foundations of analysis and definition of the structure of the set $\mathbb R$ of real numbers

*point set topology

*set theory.
A: Well we have something called the order topology which we define on a totally ordered set. It can be thought of as the generalization of the standard topology given on $\Bbb{R}$.
You can read about order topology here : http://mathworld.wolfram.com/OrderTopology.html
A: In high school you learn to work with the system ${\mathbb Q}$ of rational numbers, or with the smaller system ${\mathbb D}$ of finite (binary or) decimal fractions. While these systems are unproblematic for daily business they are unsatisfactory from a mathematical standpoint: Many important numbers that we know should be there, are missing; e.g., ${1\over3}\notin{\mathbb D}$, $\sqrt{2}\notin{\mathbb Q}$, etcetera.
There are various "bigtime" abstract constructions that fill in these holes, so that a homogeneous continuum of real numbers ${\mathbb R}$ results.
One such construction uses "Dedekind cuts" and produces a system ${\mathbb R}_{\rm Ded}$ that is so called order complete. Another construction uses "Cauchy sequences" and produces a system ${\mathbb R}_{\rm Cau}$ that is  metrically complete.
Whichever approach you take, you can prove at the end that your ${\mathbb R}_{\rm Ded}$ is in fact also metrically complete, resp. that your ${\mathbb R}_{\rm Cau}$ is in fact also order complete – the reason being that  ${\mathbb R}_{\rm Ded}$ and ${\mathbb R}_{\rm Cau}$ are "isomorphic".
