Note that these properties are also satisfied if "closure", "superset", "intersection", "contains/containing", "smallest" and "closed" are replaced by "interior", "subset", "union", "contained in", "largest", and "open". For more on this matter, see closure operator below.
Consider a sphere in 3 dimensions. Implicitly there are two regions of interest created by this sphere; the sphere itself and its interior (which is called an open 3-ball). It is useful to be able to distinguish between the interior of 3-ball and the surface, so we distinguish between the open 3-ball, and the closed 3-ball – the closure of the 3-ball. The closure of the open 3-ball is the open 3-ball plus the surface.
On the set of real numbers one can put other topologies rather than the standard one.
These examples show that the closure of a set depends upon the topology of the underlying space. The last two examples are special cases of the following.
In general, the closure operator does not commute with intersections. However, in a complete metric space the following result does hold:
One may elegantly define the closure operator in terms of universal arrows, as follows.