138 Chapter 3 • Topology of the Real Number System
analysis. Indeed, every aspiring mathematician should eventually take a course
in the subject. In this text, we present only enough of the fundamental ideas
to get us through elementary analysis. For further knowledge or study, the
reader is encouraged to consult any general topology textbook listed in the
Bibliography. Especially good introductions can be found in [4], [36], [70], and
[94]; for more challenging presentations start with [98] or [137].
3.1 Neighborhoods and Open Sets
We begin with the fundamental concept of neighborhood, which leads directly
to the concept of open set. While our context is the real number system, which
is one dimensional, these ideas derive their power from. the ease with which
they generalize to higher dimensions. Indeed, your instructor may choose to
illustrate each of the following ideas with two-or three-dimensional drawings.
Definition 3.1.1 Let x E IR and c > 0. The interval ( x -c, x + c) will be called
thee-neighborhood of x and denoted N 0 (x). Geometrically speaking, N 0 (x)
is the set of all points that are within a distance of c from x.
x-e x x+e
Figure 3.1We often say simply "neighborhood^1 of x," by which we will always mean
"t:-neighborhood of x , for some c > O.'' We shall see that the language of
neighborhoods is quite useful in expressing concepts of analysis.
Examples 3.1.2 Uses of the language of neighborhoods:
(a) A sequence {xn} converges to Liff 'r:/g > 0, Xn is eventually in N 0 (L). In
words, a sequence converges to L iff it is eventually in every neighborhood
of L.(b) A sequence { Xn} has a subsequence converging to L iff '<le > 0, Xn is
frequently in N 0 (L). In words, a sequence has a subsequence converging
to L iff it is frequently in every neighborhood of L.- In topology, the term "neighborhood" has a slightly more general definition.