Order for sets in the real line

Question

Consider the sets $[0,1]$ and $[1,2]$.

I want to say that $[1,2]$ is greater than $[0,1]$. Is there a set order such that

$$A \geq B \quad \text{if} \quad \inf A \geq \sup B.$$

What is the name of such order?

Answer 1

If a relation is labeled as an "order" then there are two possibilities: it is irreflexive or it is reflexive.

If $A$ is a singleton then $A\geq A$ so the relation is not irreflexive.

If $A$ has two distinct elements then $A\not\geq A$ so the relation is not reflexive.

Conclusion: we are not dealing with a relation that gets the label "order" in mathematics.