# Closed set

### From Glossary

A set is closed if it contains all its limit points, i.e. if
and
each is in , then is in . By the logic of the definition,
an empty set is closed (it is also open). A finite number of
intersections of closed sets is a closed set, so the feasibility region of a
mathematical program in standard form is closed if
each constraint function is continuous. ( can be
only upper semi-continuous for the constraint,
.) An example of a set that is not closed is: . The point 0 can be
approached by a sequence, say , but 0 is not in the set.
The *closure* of a set is the union of the set and all of its
limit points.