Prove that a straight line in the complex plane is a connected set.

I have come up with the following:

By the definition of connected sets, the line would have to contain two disjoint open sets, and by the definition of open set, a line would have to contain a neighborhood centered at z with radius r. But by definition of a line, it does not contain a neighborhood and, therefore, cannot be open and must be closed. So it must also be connected.

Does that make sense?