Originally Posted by

**My Little Pony** **Problem**

Show that an open set $\displaystyle \Omega \in \field{C}$ is pathwise connected if and only if it is connected.

**Attempt at Solution**

Suppose $\displaystyle \Omega$ is not connected and suppose $\displaystyle \Omega$ is pathwise connected. Then, for any points $\displaystyle a,b \in \Omega$ there exists a continuous function $\displaystyle f$ which connects $\displaystyle a$ and $\displaystyle b$ together. Let $\displaystyle V,W \subset \Omega$, where $\displaystyle V \cap W = \emptyset$ and $\displaystyle \Omega = V \cup W$. Let $\displaystyle a \in V$ and $\displaystyle b \in W$. Clearly, we have a contradiction because there cannot exist a continuous function from $\displaystyle a$ to $\displaystyle b$ if $\displaystyle \Omega$ is disconnected. Thus $\displaystyle \Omega$ must be connected.

How would I prove the other direction? Is this correct, btw?