How long is a wound considered open?

I've been having this debate with someone who is a bit of a germaphobe in my view.

Say I have a paper cut on my finger. It's deep enough to bleed initially, but it stops within a couple minutes. Maybe I need a bandaid at first, but after a day or 2, it is past the point where it starts bleeding again if I accidentally hit it or rub it on something. A couple days on, it still has the little flap of skin on it. Later, that goes away, but you can still see where it was. Finally, there is no trace. At what point in this process is the wound effectively closed?

When can you resume normal activities (dish washing, food prep, go in a pool/hot tub, have a dog lick your hand) and not worry about infection?