I need to determine whether the following is true or false: Given two disjoint non-empty open sets, could their union be path-connected?
I know the answer is no, but I'm struggling with the proof. I thought about assuming there is a path, and getting a contradiction somehow, but I dont really know how to do it.
Edit The sets are not empty.
Yes, the union can be path connected: let one of the two sets be empty.
If you add the stipulation that one of the sets is non-empty, then the answer is no. Since the union is the union of two non-empty, disconnected, open sets, the union is disconnected. Path connected implies connected, so by the contrapositive, this space must not be path connected.
To go from the definition to this proof, all you have to show is Path Connected -> Connected. To do this: Show that if $A$ and $B$ are connected and have a point in common, then $A \cup B$ is connected. Next, show that the interval $[0, 1]$ is connected (the hardest part, but not overly challenging). Next, the image of any connected space under a continuous map is connected.
Now to put this all together, let $A$ be a path connected space, and let $x \in A$. For all $y \in A$, there exists a path $f_y$ such that $f_y(0) = x$ and $f_y(1) = y$. Then, $A = \bigcup_{y \in A} f_y([0, 1])$. Each of $f_y([0, 1])$ are connected, and they all have $f_y(0) = x$ in common, so $A$ is connected.