The union and intersection of sets can be extended to the union and intersection of many sets, even an infinite number. When dealing with a collection of several sets, it is usual practice to refer to them as families or classes of sets. To denote a family of sets, one often uses indices such as {A1, A2, A3, … , A10} and for an infinite family, we might write {A1, A2, A3, …} or
Other common ways to denote families of sets are
where the set Λ is called an index set.
For example
The reader might recall the notation for infinite sums and products as
which motivates the following notation and definition for sets.
The following examples illustrate these ideas.
Many rules for the intersection and union of sets introduced in Section 2.1 can easily be extended to families of sets. We leave the proofs to many of these laws to the reader.
Proof of d)
(⊆): To show we let
Hence,
(⊇): The proof of the set containment ⊇ as well as the proofs of a), b), and c) follows along similar lines and is left to the reader.
A topology on a set is a family of subsets of the set that places a “structure” on the set that allows for the study of convergence and limits of points in the set. The study of point‐set topology, which we introduce in Section 5.4, forms the “metrical” structure for several areas of mathematics, such as real and complex analysis.
The idea is to introduce a family J of subsets of a given set U, such as a family J of subsets of the real numbers U = ℝ. The sets in the family J are called open sets, and these sets act as “neighborhoods” of points in U, allowing for the discussion of convergence sequences in U. The family of open sets J is called a topology on U. But not any collection of subsets of U is a topology. There are three restrictions on a family J in order that it be a topology on U, They are as follows:
Let A1 = {1, 2}, A2 = {2, 3}, A3 = {3, 4} and in general Ak = {k, k + 1}. Write explicitly the following sets:
Find the infinite union and intersections
of the following sets:
Define a family of subsets
of the plane ℝ2 by where m, n ∈ ℕ. Find the following sets. Hint: Proceed like one does with double series.
Prove the following distributive property for indexed families of sets:
Let A be a set and ℑ a collection of subsets of A. The collection ℑ is called an algebra1 of sets if
When this happens, we say the family ℑ is closed under unions and complementation. Which of the following collections of subsets of A = {a, b, c} constitute an algebra of subsets of A ?
In measure theory, a subset A of the real numbers is said to have length (or measure) zero if ∀ε > 0 and there exists a sequence of intervals Ak = (ak, bk) that satisfy
where their total length is less than ε; that is
Show that any sequence of real numbers {ck, k = 1, 2, …} has measure zero. Hint: Cover {ck} by a union of intervals (ak, bk), where the length of each interval satisfies |bk − ak| = ε/2k. Use this result to show that the rational numbers have a total length of 0.
A subset A of the real numbers is said to be compact if for every collection ℑ = {(aα, bα) : α ∈ Λ} of open intervals whose union contains (or covers) A; i.e.
there exists a finite subcollection of intervals of ℑ whose union also contains (or covers) A. Show the set A = (0, 1) is not compact by showing the following.
Verify that the following families of subsets of A form topologies on A = {a, b, c}.
Which of the families of subsets of {a, b, c} are topologies on {a, b, c}?
Find an infinite family of sets whose intersection is
Find an infinite family of sets whose union is
There is a wealth of information related to topics introduced in this section just waiting for curious minds. Try aiming your favorite search engine toward unions and intersections of families of sets, topology on a finite set.
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