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**HAL9000** 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 $\displaystyle $X$ $ is a collection $\displaystyle \mathcal{T}\subseteq\mathbb{P}(X)$ of subsets of $\displaystyle $X$$, called open sets, such that the empty set and the whole of $\displaystyle $X$$ are in $\displaystyle $X$$, and $\displaystyle \mathcal{T}$ is closed under finite intersections and arbitrary unions. In symbols:

(1) $\displaystyle $\emptyset,X\in\mathcal{T}$$;

(2) If $\displaystyle $O_i\in\mathcal{T},i\in{I}$$, where $I$ is a finite index set, then $\displaystyle $\bigcap_{i\in{I}}O_i\in\mathcal{T}$$;

(3) If $\displaystyle $O_j\in\mathcal{T},j\in{J}$$, where $J$ is any (finite or infinite) index set, then $\displaystyle $\bigcup_{j\in{J}}O_{j}\in\mathcal{T}$$.

And now it is said that:

The defining property (1) may be dropped. In fact, $\displaystyle $\emptyset \in \mathcal{T}$$ follows, if we apply **(3)** to the empty family, and $\displaystyle $X \in \mathcal{T}$$ 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, $\displaystyle $\emptyset \in \mathcal{T}$$ follows, if we apply **(2)** to the empty family, and $\displaystyle $X \in \mathcal{T}$$ 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?