Closure

A set $S$ is closed under an operation $\ast$ if applying the operation to elements of $S$ always produces an element of $S$:

Examples:

• $\mathbb{N}$ is closed under addition and multiplication, but not subtraction ($1 - 2 \notin \mathbb{N}$)
• $\mathbb{Z}$ is closed under $+, -, \times$ but not division
• $\mathbb{Q}, \mathbb{R}, \mathbb{C}$ are closed under all four arithmetic operations (excluding division by zero)
• The set of even integers is closed under addition; odd integers are not ($1 + 1 = 2$)

Closure of a set (noun): given $S \subseteq U$ and an operation $\ast$, the closure $\overline{S}$ is the smallest superset of $S$ that is closed under $\ast$. Constructively: keep applying $\ast$ to elements until no new elements appear.

Closure of relations: given a relation $R$ on $X$, extend $R$ minimally to satisfy a property.

• Reflexive closure: $R \cup \{(x,x) : x \in X\}$
• Symmetric closure: $R \cup R^{-1}$
• Transitive closure $R^+$: smallest transitive relation containing $R$ — if $(a,b), (b,c) \in R^+$ then $(a,c) \in R^+$. Computed by repeated composition: $R^+ = \bigcup_{n \geq 1} R^n$

Where closure shows up:

• Algebra: groups, rings, and fields are defined by closure axioms — a group $(G, \ast)$ requires $G$ closed under $\ast$
• Linear algebra: a subspace must be closed under addition and scalar multiplication
• Measure theory: a $\sigma$-algebra is a collection of sets closed under countable unions and complements — foundational for probability
• Topology: the topological closure $\overline{A}$ is the smallest closed set containing $A$ (contains all its limit points)
• Formal languages: the Kleene star $L^*$ is the closure of a language $L$ under concatenation
• Databases / graphs: transitive closure answers reachability — "can I get from $a$ to $b$?"

Intuition: closure is the answer to "if I start with this set and keep doing this operation, what's the smallest set I can't escape?" It's the fixpoint of "add everything forced to be here."

See also: Vector Spaces and Basis, Graphs and Traversals