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Empty family of sets

  1. Jul 25, 2009 #1
    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.
     
  2. jcsd
  3. Jul 25, 2009 #2

    CompuChip

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    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).
     
  4. Jul 25, 2009 #3
    Do you mean that "for all" incorporates "for no"?
     
  5. Jul 25, 2009 #4

    CompuChip

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    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.
     
  6. Jul 27, 2009 #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
     
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