- #1
blahblah8724
- 32
- 0
For a subset which is both closed and open (clopen) does its closure equal its interior?
A clopen set is a set that is both closed and open. This means that the set contains all of its limit points and does not contain any of its boundary points.
The closure of a set is the smallest closed set that contains all the points of the original set. It includes all the limit points of the set as well as the boundary points.
No, a set cannot be both open and closed. If a set is open, it does not contain any of its boundary points, whereas a closed set contains all of its boundary points. Therefore, a set cannot be both open and closed at the same time.
A set is clopen if it is both open and closed. This can be determined by checking if the set contains all of its limit points and does not contain any of its boundary points.
The closure of a set includes both the interior and the boundary points of the set. This means that the interior of a set is a subset of its closure. In other words, the closure is the union of the interior and the boundary of a set.