Question

Consider RxR with dictionary order. Let us call an interval in this ordered set vertical if its endpoints have the same x coordinate. That is, a vertical interval has the form (axb,a,d) for real numbers a,b,d. Show that the collection of all vertical intervals is a basis for the order topology on RxR.

Please explain the question. What is the objective and enough to show.
Please walk me through. I have been trying to figure out the question for a while.

Answer 1

I didn't know what the notation (axb,a,d) means, so I'm going to follow the easy notation used in Munkres (not sure). Let me know in the comments if any of the steps seem unnatural to you. Also note that the set in the end is valid for a<x<b. For the end point we argue that the copy of R is of the form {a}x(b,infinity) & {c}x(-infinity,d)