I got somewhat stuck when I read that in the definition of a topology on a set one may drop the first defining property.
The familiar definition reads as follows:
A Topology on a set is a collection of subsets of , called open sets, such that the empty set and the whole of are in , and is closed under finite intersections and arbitrary unions. In symbols:
(2) If , where $I$ is a finite index set, then ;
(3) If , where $J$ is any (finite or infinite) index set, then .
And now it is said that:
The defining property (1) may be dropped. In fact, follows, if we apply (3) to the empty family, and follows, if we apply (2) to the empty family.
Now I do see that we have to do with an instance of vacuous truth here, because whenever the index set is empty we have nothing to check.
But on this ground I could have swapped the roles, saying that:
The defining property (1) may be dropped. In fact, follows, if we apply (2) to the empty family, and follows, if we apply (3) to the empty family.
Am I wrong? Is there possibly more to this than simply addressing to vacuous truth?