Is union of disjoint path-connected sets not path-connected? Is the following true:

The union of disjoint path-connected sets is not path-connected.

 A: You have to require that the sets are open and non-empty to make the statement interresting (otherwise we could just take one path-connected set and partition it and disprove the statement).
Assuming that makes the statement equivalent to if path-connectedness implies connectednes for open sets, and yes that's true.
Assume that an open set is path-connected and assume we can decompose it as union of two disjoint open non-empty sets $\Omega_1$ and $\Omega_2$. Then we can select two points $x_1\in\Omega_1$ and $x_2\in\Omega_2$ (since they're non-empty).
Now consider a path $\gamma$ from $x_1$ to $x_2$ that is a continuous function from $[0,1]$ to the set and $\gamma(0)=x_1$ and $\gamma(1)=x_2$. Now we consider $t=\sup_{\gamma(\tau)\in\Omega_1}\tau$. Now for an neighborhood $V$ of $\gamma(\tau)$ there should be an neigborhood of $\tau$ such that $\gamma(t)\in V$ in that neighborhood. But as any neigborhood of $\tau$ contains points that maps to both $\Omega_1$ and $\Omega_2$ we have a contradiction.
