Final answer:
The entire xy-plane is open but not closed because it has no boundary and, therefore, does not include a boundary. An open set does not include its boundary and a closed set contains all its limit points, which the entire plane lacks.
Step-by-step explanation:
The statement is true: the entire xy-plane (our usual x-y plane) is an example of a set in the plane that is open but not closed.
In topology, an open set is a set that does not include its boundary. In the context of the xy-plane, the 'entire' plane has no boundary, so every point within it does not have a neighboring point that is 'outside' since there is no 'outside.' Hence, it is considered open.
Conversely, a closed set would include its boundary. Since the xy-plane lacks a boundary, it cannot be closed. In addition, for a set to be closed, it must contain all its limit points, and since any 'outside' limit points do not exist for the entire plane, it cannot be closed.