Problem

Show that an open set is pathwise connected if and only if it is connected.

Attempt at Solution

Suppose is not connected and suppose is pathwise connected. Then, for any points there exists a continuous function which connects and together. Let , where and . Let and . Clearly, we have a contradiction because there cannot exist a continuous function from to if is disconnected. Thus must be connected.

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