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 $$S$$.