There are two meanings of the word "closed".
In topology, a region is "closed" iff its complement is open, or alternatively iff it contains all its limit points.
Some examples of closed regions are rectangles with boundary and circles with boundary.
We may also call a manifold "closed" iff it has no boundary, yet is compact.
A set is closed under a function iff (where is the number of arguments that accepts - possibly one).
For example, the real numbers are closed under addition.