So I have Vector space V over the field F. And we have two subspaces U,W (subspaces of V)

There is a claim here saying that the intersection of the subspaces U and W is a vector space too, but their unity is not a subspace.

How come their unity is not a subspace? Shouldn't their unity be equal to the subspace U+W since that it's going to be closed under addition and multiplication with scalars? What am I missing here?