Definition:Closure (Abstract Algebra)

Algebraic Structures
Let $$\left({S, \circ}\right)$$ be an algebraic structure.

$$S$$ is defined as being closed under $$\circ$$, or $$\left({S, \circ}\right)$$ is closed iff:

$$\forall \left({x, y}\right) \in S \times S: x \circ y \in S$$

Some authors use stable under $$\circ$$.

Scalar Product
Let $$\left({S: \circ}\right)_R$$ be an $R$-algebraic structure.

Let $$T \subseteq S$$ such that $$\forall \lambda \in R: \forall x \in T: \lambda \circ x \in T$$.

Then $$T$$ is closed for scalar product.

If $$T$$ is also closed for operations on $$S$$, then it is called a closed subset of $$S$$.

Topology
Let $$X$$ be a topological space, and let $$A \subseteq X$$.

Then the closure of $$A$$, denoted by $$\overline{A}$$ or $$\operatorname{cl}(A)$$, is defined as
 * $$\overline{A} := \bigcap_{A \subset B \subset X, B \text{ closed}} B$$.

Equivalently,
 * $$\overline{A}$$ is the smallest closed set that contains $$A$$;
 * $$\overline{A}$$ is the union of $$A$$ and its boundary;
 * $$\overline{A}$$ is the union of $$A$$ and its limit points;
 * $$\overline{A}$$ is the union of all isolated points of $$A$$ and all limit points of $$A$$.