In general, we say that a set is closed under an operation if applying the operation to any members of the set results in a value that is also a member of the set. Thus we say that the set of natural numbers is closed under addition and multiplication, but not under subtraction or division (since for instance 3÷2 is not a natural number, or even an integer).
We say that an interval of the real number line is closed if it contains its endpoints. This is indicated in interval notation by enclosing the coordinates in square brackets. Thus, the interval \(0 \leq x \leq 2\) would be denoted by \([0,2]\).
The word closed may have much more specialized meanings depending on context.
A closed curve in the plane is a continuous curve without endpoints that completely encloses a region of the plane. A simple closed curve is a closed curve that does not cross itself.
A subset of a metric space is closed if it contains all of its accumulation points, or equivalently if every convergent sequence in the subset is a Cauchy sequence.
A set in a topology is closed if its complement is an open set.
- [MLA] “closed.” Platonic Realms Interactive Mathematics Encyclopedia. Platonic Realms, 25 Mar 2013. Web. 25 Mar 2013. <http://platonicrealms.com/>
- [APA] closed (25 Mar 2013). Retrieved 25 Mar 2013 from the Platonic Realms Interactive Mathematics Encyclopedia: http://platonicrealms.com/encyclopedia/closed/