There are two meanings of the word "closed".

In topology

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.

In functions

A set $S$ is closed under a function $f$ iff $x_1, ... x_t \in S \implies f(x_1, ... x_t) \in S$ (where $t$ is the number of arguments that $f$ accepts - possibly one).

For example, the real numbers are closed under addition.

See also