An indexed family is a family (set of sets), but such that its elements (sets) are indexed (e.g. by natural numbers). Formally an indexed family is a function from the set of indices into the set of its elements.
E.g. is a family of intervals of the real line, indexed by natural numbers.
Unions and intersections of indexed families
We will be especially interested in unions and intersections of indexed families. We calculate those unions and intersection similarly to the case of simple families of sets, but this time we can include information which indices we would like to use.
For the family in the above example: , because it is clear that no negative number is in this union. On the other hand, if is a real number, then let . Then , and so .
On the other hand , because for all , and .
Meanwhile, e.g. , and .
Double indexed families
Obviously the set of indices can be a product of two sets, so sets in the family can be indexed by two indices. E.g.: , .
Therefore we can imagine such a family as an infinite array of sets:
Double unions and intersections
Given a double indexed family, the expressions of the following type make sense: e.g. , czy and other similar. Let us calculate those two expressions.
To calculate the first expression, denote — it is the intersection of all the sets in the -th column of this infinite array. It is clear that , because that for any , and , so . Therefore, , because elements of all the sets in this union are less than or equal to zero and for all , for .
To calculate the second expression, denote , so it is the union of all sets in the -th row starting with the -th element. So , because for any , and if , then . On the other hand if , then , for . Therefore, . That is because for any , and , and therefore