Why Read This Book? - Index of

178 Chapter 5 The Real Numbers
The next exercise gives us a way to think about a closed set without having to
refer to its complement.
EXERCISE 5.4.12 A set is closed if and only if it contains all its cluster points.
EXERCISE 5.4.13 Show that {1/n} ∞ n=1 is not closed.
5.5 Closure of Sets
If a set A is not closed, we might want to close it off, so to speak, by finding the
smallest closed superset of A, if there is one. First we define a term for this smallest
closed superset of A. Then we address whether it exists, and if so, whether it is
unique.
Definition 5.5.1 Suppose A is a set of real numbers, and suppose C is a set with
the following properties.
(C1) A ⊆ C.
(C2) C is closed.
(C3) If D is a closed set and A ⊆ D, then C ⊆ D.
Then C is called a closure of A, and is denoted A.
Notice how Definition 5.5.1 lays down the characteristics that the set C must
have in order for it to qualify as a smallest closed superset of A. Property C1
guarantees that any set we would call A is indeed a superset of A. Property C2
guarantees that it is closed, and property C3 says that no closed superset of A will
be any smaller.
Let A be a set, and consider the family of subsets of the real numbers, where
each set in the family is a closed superset of A. That this family is non-empty is
immediate, for R itself is a closed superset of A. In the next exercise, you will show
that the closure of a set exists uniquely. What you will show is that the intersection
of all closed supersets of A has properties C1–C3 and that any two sets that both
have properties C1–C3 are actually the same.
EXERCISE 5.5.2 Suppose A is a set of real numbers. Then A exists uniquely
and can be constructed as
A = �
S 17
(5.11)
S⊇A
S closed
17 See Exercises 3.3.14 and 5.3.11 and Theorem 3.3.11.