During AMC testing, the AoPS Wiki is in read-only mode. No edits can be made.

Closed

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

Invalid username
Login to AoPS