154 Chapter 3 • Topology of the Real Number System
- (Project) In this exercise we shall denote the closure of a set A by Ac!
rather than by A. Start with a set A. Then perform the operations of
complement and closure, alternating in succession, forming the sequences
A, Ac, Acc1, A cclc, Acc1cc1,... , and A, Ac!, Ac1c, Ac1cc1, Ac1cc1c, ....
3.3
(a) Prove that this process can never yield more than 14 different sets.
(b) Find a set A for which this process yields exactly 14 different sets.- Compact Sets
Instructors may omit this entire section
and replace it by the following definition:Definition: A set of real numbers is compact if it is closed and bounded.
We sh all not need to use the concept of compactness until Section 5.3. When
we do use the concept, the above definition will suffice. Instructors who choose
to cover the remainder of this section will do so out of a desire to place the
concept of compactness upon a solid topological foundation. Please disregard
the above definition if you are studying this section.
TOPOLOGICAL TERMS
Definition 3.3.1 A topological term or concept^3 is a term or concept
definable using only the terminology of sets and open sets. Some topological
terms are:
Term:
open
closed
interior
exterior
boundary
isolat ed
cluster
dense in
closure ofX ->
Attribute of:
a set
a set
a point, relative to a set
a point, relative to a set
a point, relative to a set
a point, relative to a set
a point, relative to a set
a set, relative to IR or another set (see Exercise 3.2.28)
a set
a sequence, and a real number (see Exercise 3.1.21)- This definition would not satisfy a topologist, but it s uffices for our purposes.
•An asterisk before a t heorem, proof, or other item in this chapter indicates tha t the item is
ch a llenging or can be omitted, especially in a one-semester course.