The definition of net in topology is defined via a directed set, that is a set $A$ with a preorder such that every finite subset of $A$ has an upper bound.
If my understanding is correct, an ordered set that is directed does not mean every infinite subset has an upper bound. I felt the latter would be more useful if Zorn's Lemma is used in proving a theorem of nets. So is there a reason for not defining this way, and does it really make a difference for all the theorems about nets?
Maybe accepting every subset has an upper bound is too strong, but why do we stick with finite ones, instead of allowing countable, or etc.