Does a vector space need to be closed? What part of the definition of a vector space (see here) requires it to be closed under addition and multiplication by a scalar in the field? I would understand if we defined a vector space as a group of vectors rather then a set but we don't, also non of the axioms require this to be a condition?

Because, addition and multiplication (by a scalar) operations are functions from $V \times V$ to $V$ and $K \times V$ to $V$, respectively.
