If we want to add one real number the simplest way to do it is to use Cohen forcing. The poset is $\lbrace p\colon n\to 2\mid n\in\omega\rbrace$ which is a countable set. We can think of this as approximating a new set by finite sets.

If we want to add $\omega$-many pairwise generic real numbers we can do it by taking the poset $\lbrace p\colon n\times m\to 2\mid n,m\in\omega\rbrace$. This forcing approximates infinitely many new sets by finite parts of finitely many of the new sets. Interestingly enough both these sets are countable and therefore forcing with one is the same as forcing with the other.

But we can also take $\lbrace p\colon\omega\times n\to 2\mid n\in\omega\rbrace$ and approximate *all* new sets at each stage. This poset is not countable anymore.

If we look at these things as topological spaces then the original Cohen forcing is really just a countable dense subset of the Cantor space, and we are adding a generic point to the Cantor set. The second forcing is the product of countably many Cantor sets, it is homeomorphic to the original Cantor set.

The third notion of forcing, though, is the box product of countably many Cantor sets. So at least topologically it is different. I also have to admit that I never saw anyone using this approach.

Is there anything wrong it? Is it at all different from the first/second approach? What do we gain/lose when we use the box-product over the usual product?