Hello!

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:

(1)

;

(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.

…couldn't I?

Am I wrong? Is there possibly more to this than simply addressing to vacuous truth?