Why Read This Book? - Index of

176 Chapter 5 The Real Numbers
Here are some immediate observations.
1. Int(A) ⊆ A. For if x ∈ Int(A), then there is an ɛ-neighborhood of x contained
entirely within A, and x is in this neighborhood. Thus x ∈ A.
2. Ext(A) ⊆ AC by similar reasoning.
3. If x ∈ Bdy(A), then x might or might not be in A. However, if x ∈ A, then every
neighborhood of x contains elements of AC . Similarly, if x/∈ A, then every
neighborhood of x contains elements of A.
4. Int(A) = Ext(AC ).
5. Ext(A) = Int(AC ).
6. Bdy(A) = Bdy(AC ).
EXERCISE 5.4.1 For each of the following sets, determine the interior, exterior,
and boundary.
(a) {1}
(b) (0, 1]
(c) {1/n} ∞ n=1
(d) ∅
(e) Z
(f) R
(g) Q
The proof of the next exercise is quick. It is useful because it gives us an
equivalent way to think about open sets.
EXERCISE 5.4.2 A set A is open if and only if Int(A) = A.
EXERCISE 5.4.3 If L is the LUB of a set A, then L ∈ Bdy(A).
5.4.2 Cluster Points
Once again, let A be a subset of the real numbers, and let x be any real number. It
might be that elements of A are packed densely around x.
Definition 5.4.4 A real number x is said to be a cluster point of A, provided
every ɛ-neighborhood of x contains a point in A other than x itself. That is, for all
ɛ>0, there exists a ∈ A ∩ Nɛ(x), where a �= x.
A cluster point of A might or might not be an element of A.