Graphics Reference
In-Depth Information
tion continuous. It is the foundation of calculus, analysis, and any sort of geometric
investigations. In this section we look only at some of the important definitions
as they apply to Euclidean space. Chapter 5 will study abstract topological spaces.
Because we shall encounter many of the same definitions there in a more general
context, we shall postpone some proofs and results to that chapter to avoid stating
and proving theorems twice. By and large, this section is simply a collection of defi-
nitions and essentially immediate consequences. By carefully asking “what does this
mean?” the reader should have little difficulty in proving most theorems.
Unless stated otherwise, all points and sets in this section belong to
R
n
for some
fixed but arbitrary n.
Definition.
A set
N
is said to be a
neighborhood
of a point
p
if it contains an open
ball about
p
, that is, there exists an e>0 such that
B
n
(
p
,e) Õ
N
.
The important part of the neighborhood
N
is that it contains some open ball about
p
. We do not care whether it contains some other “junk.” For example, the open inter-
val (-1,1) together with the set {-100,23,5} would be called a neighborhood of the
origin in
R
. Note also that this definition and many others depend on the dimension
of the Euclidean space with respect to which we are making the definition. For
instance, the interval (-1,1) is a neighborhood of the origin for
R
but
not
for
R
2
. The
open unit ball
B
2
, on the other hand,
is
a neighborhood of the origin for
R
2
.
Definition.
A subset
X
of
R
n
is called an
open set
if for all points
p
in
X
there is
an e>0 such that
B
n
(
p
,e) Õ
X
.
X
is said to be a
closed set
if the complement of
X
,
R
n
-
X
, is open.
Specifying the open sets of a set is what defines its “topology.” (A precise
definition of the term “topology” is given in Chapter 5). “Open” and “closed” are dual
concepts. The open interval (0,1) is an open set in
R
and the closed interval [0,1] is a
closed set. Our definitions are compatible with the old usage of the terms. A set does
not have to be either open or closed. For example, the “half-open” (“half-closed”) inter-
val (0,1] is neither. The set
R
n
is
both
open and closed as is the empty set f but these
are the only subsets of
R
n
that are both open and closed. A single point is always a
closed set. In practice, sets tend to be open if the “<” relation is used in their defini-
tion and closed if the “£” relation is used.
4.2.1. Proposition
(1) The arbitrary union of open sets is an open set.
(2) A finite intersection of open sets is open.
(3) The arbitrary intersection of closed sets is a closed set.
(4) A finite union of closed sets is closed.
Proof.
Part (1) is easy. To prove (2), let
O
1
,
O
2
,...,
O
k
be open sets and let
p
be a
point in their intersection. Since each
O
i
is an open set, there exists an e
i
> 0, so that
B
n
(
p
,e
i
) Õ
O
i
. If
{
}
e
=
min
e
,
e
,...,
e
,
12
k
