Which property is not required for closure under addition in a vector space?

Closure under addition in a vector space requires that the sum of any two vectors within that space also results in a vector that remains in that same space. This is fundamental because it ensures that the vector space is stable under the operation of addition.

The requirement that the sum can be performed in any order refers to the commutative property of addition, which is typically accepted but doesn't directly impact the closure property itself. The statement that the sum remains in the same space emphasizes the definition of closure.

In the context of vector spaces, the notion that the result of adding two vectors must lie outside the set contradicts the very definition of closure under addition. Hence, this property is not required for the concept of closure, making it the correct answer. This reinforces the idea that operations involving vectors in a vector space must yield results that remain within that space, maintaining its structure and properties.