Graphics Reference
In-Depth Information
Definition.
Let
A
be a subset of a topological space
X
. A
boundary point
of
A
is a
point
p
in
X
such that every neighborhood of
p
meets both
A
and the complement of
A
,
X
-
A
. The
boundary
of
A
(in
X
), denoted by bdry(
A
), is defined to be the set of
boundary points of
A
. A point
a
in
A
is called an
interior point
of
A
(in
X
) if
a
has a
neighborhood
U
in
X
that is contained in
A
. The set of interior points of
A
is called
the
interior
of
A
and is denoted by int(
A
). The
closure
of
A
(in
X
), denoted by cls(
A
),
is defined to be the set of all points
p
in
X
with the property that every neighborhood
of
p
meets
A
.
These definitions are analogous to those in Section 4.2. It is important to note
that the above definitions of boundary, interior, and closure are all
relative
to the con-
taining space
X
. One can show (Exercise 5.3.2) that
()
=-
()
int
AA
bdry
A
,
(5.6)
()
=
()
»
()
cls
A
int
A
bdry
A
.
(5.7)
Definition.
A subset
A
of a topological space
X
is said to be
dense
in
X
if it inter-
sects every nonempty open set of
X
. It is said to be
nowhere dense
in
X
if its closure
contains no nonempty open subset of
X
.
For example, the rational numbers are dense in
R
, as are the irrational numbers.
Any finite subset of
R
n
is nowhere dense in
R
n
.
5.3.7. Lemma.
Let (
X
,T) be a topological space and let
Y
be a subset of
X
. Define
a set S of subsets of
Y
by
{
}
S =«
AYAT
.
Œ
(5.8)
Then S is a topology on
Y
.
Proof.
The proof is straightforward.
Definition.
Let (
X
,T) be a topological space and let
Y
be a subset of
X
. The topol-
ogy S on
Y
defined by equation (5.8) is called the
relative topology on
Y
induced by T
.
The topological space (
Y
,S) is called a
subspace of (
X
,T)
. If T is clear from the context,
one simply talks about the
subspace
Y
of
X
.
“Induced metrics” and “relative topologies” may sound abstract, but these
notions tend to get used automatically in the context of subspaces of
R
n
, although
without explicitly using those terms. For example, one probably had no hesitation to
talk about the distance between points
of a circle
or open arcs
in a circle
.
5.3.8. Lemma.
Let
X
be a topological space and let
Y
be a subspace of
X
. The closed
sets of
Y
are just the sets
A
«
Y
, where
A
is closed in
X
.
Proof.
This is an easy consequence of the definitions.