This is just the Cartesian product of two copies of . So as a vector space over it is clearly not closed under scalar multiplication. Let the set of real numbers that are not integers. Just consider . You may notice if you multiply instead by a scalar that is an integer, your product will stay in W as the integers are closed under multiplication.
If you are interested, you can consider it as something of a vector space over . These are called modules, and this W is in fact an example of a two dimensional free module with basis .