Empty Family of Sets: Does it Make Sense?

[littleHilbert]

Hi! I'd like to ask the following question.

Does it make sense to take unions and intersections over an empty set?

For instance I came across a definition of a topological space which uses just two axioms:

A topology on a set X is a subset T of the power set of X, which satisfies:
1. The union of any familiy of sets in T belongs to T. Applying this to the empty family, we obtain in particular $$\emptyset\in{}T$$
2. The intersection of any finite family of sets in T belongs to T. Applying this to the empty family, we obtain in particular $$X\in{}T$$

The empty family is just a family of sets with an empty index set, isn't it? Or did I misunderstand the notion of the empty family.

 

[CompuChip]

The empty set is always tricky, usually statements are a matter of definition.
If you define
$$\bigcup_{i \in I} X_i$$
as
$$\{ x \in X \mid \exists i \in I: x \in X_i \}$$
and
$$\bigcap_{i \in I} X_i$$
as
$$\{ x \in X \mid \forall i \in I: x \in X_i \}$$
then the first statement is "vacuously false" (i.e. for any x, there does not exist such i in I because I is empty) and the second is vacuously true (P is always true if the index set I in "for all i in I, P holds" is empty).

 

[littleHilbert]

and the second is vacuously true (P is always true if the index set I in "for all i in I, P holds" is empty).

Do you mean that "for all" incorporates "for no"?

 

[CompuChip]

No, I am talking about vacuous truth: in mathematics, any statement of the form
$$\forall x \in \emptyset, P(x)$$
is logically true. An example in "ordinary" language is: "all white crows have three legs," which is true by the fact that there are no white crows.

Similarly here, for any x in X, the statement $\forall i \in I, x \in X_i$ is (vacuously) true, because there are no i in I.

 

[n!kofeyn]

I think that the definition littleHilbert has posted is awkward, and you're right to be confused. It doesn't read well to me, which is a quality a definition shouldn't have. The definition I've seen and like is:

T is a topology for X if it is a collection of subsets of X that satisfies:
1) the empty set and X are in T
2) T is closed under arbitrary unions
3) T is closed under finite intersections