Suppose $S = (a,b)$ and $T = (c,d)$, where:
- $a < d$.
- Both $S$ and $T$ are non-empty.
- $S \cap T = \emptyset$.
- $S \cup T$ is connected.
Now the question is: What is $b - c$?
The paradox is that, it can't be $b - c \leq 0$, because then $b$ is not included in $S \cup T$, so it's not connected. But it can neither hold that $b - c > 0$. Because if we then set $\epsilon = b - c$, $c + \epsilon / 2$ is included in both $S$ and $T$, so $S \cap T \neq \emptyset$.
Exculpate myself from disdain: Maybe at first glance everyone regards this as a trivial problem. Yes, it is, somehow. Because I'm not majoring in math and these are just random thoughts. I just want to examine what will happen when you put two open sets on the real axis close enough, and what's the critical point where they merge into a single, bigger one. All these thoughts are based on pure, plain intuition. I don't know too much definitions/axioms/theorems about real analysis. I never had that course.
Perhaps that's why this problem looks stupid. Unfortunately, I finally found the idea I wanted to express is just how connectedness is defined (also pointed out by one of the answers below). So there is no chance to win, as definition cannot be violated. But I think the idea and motivation in it is clear to everyone, and you don't need a definition to think of it. The mystery is still there: You move two disjoint open intervals closer and closer, and finally they intersect. But you don't know where! The definition itself doesn't answer it. Also, you cannot have too many definitions, because the more definitions, the more likely there is hidden inconsistency amongst them. So this problem is solved in terms of math, and I don't think anyone can solve it in terms of logic and philosophy. So I'll mark this problem as answered. Thanks for all the answers. Very helpful, indeed.