Consider a set \(S\) of elements under some operation, say \(\cdot\). We say that \(S\) is closed under \(\cdot\) if applying \(\cdot\) to the element(s) of \(S\) always results in an element of \(S\).

For example, the set of natural numbers is closed under addition but not closed under subtraction.

• complement
• union
• intersection
• set difference
• symmetric difference
• concatenation
• Kleene closure
• positive closure
• reversal
• homomorphism
• inverse homomorphism
• etc.