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true or false: the entire plane (our usual x-y) plane is an example of a set in the plane that is open but not close

User Zxz
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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.

User Michael Kreutz
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