Closed vector space, open versus closed set

Question

I read from Mathworld that; a vector space $V$ is a set that is closed under finite vector addition and scalar multiplication.

Here my understanding of the word closed is a bit ambiguous, so what does it really mean for a vector space to be closed?

If I remove the word vector space, and instead use the word set, and ask what does it mean for a set to be closed, wouldn't that be more accurate to ask?

Then what is the difference between an open set and a closed set?

Could I also remove the terms; finite vector addition and scalar multiplication in the definition? Because where is subtraction and division? Would'nt leaving out those - if the set is in $\mathbb{N}$ or $\mathbb{Z}$ be totally wrong?

Answer 1

A set $S$ is closed under an operation $*$ if $a*b \in S$ whenever $a,b \in S$. This concept has nothing to do with the concept of a closed set in a topological space. When you say 'closed under' some operation the word 'under' is very important.

Answer 2

Indeed you’re getting confused with topology notions.

To be closed under finite vector addition means that if you add a finite number of vectors of a vector space $V$, you get an element (a vector) of $V$.

This is a totally different notion to an open set or closed set in topology.