Is a totally ordered set consisting of discrete point sets containing point p a well-ordered set?

Question

I have encountered a fully ordered set and want to determine whether it is well ordered. Each element B in this totally ordered set A is a set and includes point p, which means that the elements in A may show the following situation:

• My current attempt: I want to verify that A is well-ordered, then I need to discuss the existence of a minimum element for any non-empty subset S. If there is a set B in S such that it contains only finite points, it can be easily verified $\bigcap _{B\in S} B\in S$ by proof by contradiction. But if the elements in S are at least listable point sets, I don't know how to describe it.

Thanks for all your help.

Answer 1

Your set $A$ need not be well ordered. Let $A= \{B((0, 0), r) \mid r \in \Bbb R^+ \}$, where $B((0, 0), r)$ is the ball of radius $r$ centered at $(0, 0)$. Then $A$ is order-isomorphic to $\Bbb R^+$, which is not well ordered.