Empty Family of Sets: Does it Make Sense?

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In summary, the conversation discusses the concept of taking unions and intersections over an empty set in mathematics. The definition of a topology on a set X and how it applies to the empty family is also mentioned. The conversation concludes with a discussion on vacuous truth in mathematics.
  • #1
littleHilbert
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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 [tex]\emptyset\in{}T[/tex]
2. The intersection of any finite family of sets in T belongs to T. Applying this to the empty family, we obtain in particular [tex]X\in{}T[/tex]

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.
 
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  • #2
The empty set is always tricky, usually statements are a matter of definition.
If you define
[tex]\bigcup_{i \in I} X_i[/tex]
as
[tex]\{ x \in X \mid \exists i \in I: x \in X_i \}[/tex]
and
[tex]\bigcap_{i \in I} X_i[/tex]
as
[tex]\{ x \in X \mid \forall i \in I: x \in X_i \}[/tex]
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).
 
  • #3
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"?
 
  • #4
No, I am talking about vacuous truth: in mathematics, any statement of the form
[tex]\forall x \in \emptyset, P(x)[/tex]
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 [itex]\forall i \in I, x \in X_i[/itex] is (vacuously) true, because there are no i in I.
 
  • #5
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
 

1. What is an empty family of sets?

An empty family of sets, also known as a null family, is a collection of sets that contains no elements. It is denoted by the symbol ∅ and is considered to be a subset of every other set.

2. Why would a family of sets be empty?

A family of sets may be empty if all the sets that are supposed to be included in the family have no elements in common. This could occur if the sets are defined in such a way that there is no overlap between them, or if all the sets are empty themselves.

3. Does an empty family of sets make sense?

Yes, an empty family of sets does make sense. It is a valid mathematical concept and has important applications in set theory and topology. It allows for the concept of a universal set, which contains all other sets, to exist.

4. What is the cardinality of an empty family of sets?

The cardinality, or size, of an empty family of sets is zero. This is because the family contains no elements, and the cardinality of a set is defined as the number of elements it contains.

5. How is an empty family of sets used in mathematics?

An empty family of sets is used in various mathematical proofs and definitions. For example, it is used in the definition of a topology, where the empty set is required to be included in every topology. It is also used in the definition of a sigma-algebra, where the empty set is a required element. In general, it helps to define the structure and properties of larger mathematical concepts.

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