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?

I think the Wikipedia definition is tacitly noting that your set is (a) a group under addition, since the operation is defined to give $v + w \in V$ if $v$ and $w$ are in $V$, and the associativity, identity and inverse conditions give you a group. Also (b), the action of the field $F$ on $V$ is closed again by definition (since if $\alpha \in F$ and $v \in V$, we have $\alpha v \in V$.)
An equivalent—and possibly clearer—definition of vector space would foreground that the space is a group under addition and is closed under the action of the field.
