[Math] How would the intersection of two uncountable sets form a countably infinite set

Question

This is based off my last question How would the intersection of two uncountable sets be finite?

Here is the problem(from Discrete Mathematics and its Applications)

The book's definition on countable

And the definition of having the same cardinality

I was able to get 11c pretty easily. What I thought was the intersection of the same uncountable set, say [1,2], that is [1,2]∩[1,2] would be [1,2], a uncountable set. Via help, I was able to get understand 11a. That is if you have two uncountable sets, say
(−∞,0]∩[0,∞), the intersection of those two sets would be that one value, zero, meaning it is finite countable. What I am struggling with is applying that same idea to 11b. What I thought of was having two intervals that didn't end quite at the same spot say (−∞,3]∩[0,∞) but the intersection of those would be [0, 3] which itself is a uncountable set. From 11a, what endpoints would you set on the intervals so that A ∩ B would be countably infinite?

Answer 1

How about $[0, 1] \bigcup \{2, 3, 4, 5, \dots \}$ and $[5, 6] \bigcup \{7, 8, 9, 10, \dots\}$?