Graphics Reference
In-Depth Information
Definition.
Let (
X
,d) be a metric space. A subset
U
of
X
is said to be
d-open
if for
every point
p
in
U
there is an r > 0 so that B
r
(
p
,d)
U
.
Note that a d-ball is a d-open set. In the case of the discrete metric d on a set,
every subset is d-open.
5.2.3. Theorem.
Let (
X
,d) be a metric space. Then
(1) Both the empty set f and the whole space
X
are d-open sets.
(2) An arbitrary union of d-open subsets of
X
is d-open.
(3) Any finite intersection of d-open subsets of
X
is d-open.
Proof.
The proof is similar to the proof of Proposition 4.2.1.
An
arbitrary
intersection of d-open sets need not be d-open. We already saw an
example that shows this for the Euclidean metric d on
R
in Section 4.2.
Definition.
Let (
X
,d) be a metric space. A subset
C
of
X
is said to be
d-closed
if
X
-
C
is d-open.
A (closed) interval [a,b] in
R
is a closed set with respect to the Euclidean metric.
More generally, any d-disk is d-closed, as is the d-sphere.
5.2.4. Theorem.
Let (
X
,d) be a metric space. Then
(1) Both the empty set f and the whole space
X
are d-closed sets.
(2) An arbitrary intersection of d-closed subsets of
X
is d-closed.
(3) Any finite union of d-closed subsets of
X
is d-closed.
Proof.
See the proof of Proposition 4.2.1.
Again, we already saw in Section 4.2 that an
arbitrary
union of d-closed sets need
not be d-closed. Another fact to note is that subsets do not have to be either d-open
or d-closed. As was pointed out before, the half-open interval [0,1) in
R
is
neither
d-
open
nor
d-closed with respect to the Euclidean metric d.
Definition.
X
. A subset
V
of
X
that contains
p
is called a
d-neighborhood of
p
if there is a d-open set
U
so that
p
Let (
X
,d) be a metric space and let
p
U
V
.
Note that d-neighborhoods need not be d-open. For example, [-1,1] is a d-
neighborhood of 0.
We shall see later in Section 5.3 that the important topological aspect of a space
is its collection of open sets. In that sense therefore, although a space can have
many different metrics, just because two metrics are different does not mean that the
“topology” that they induce on the space is different.
Definition.
Let d and d¢ be two metrics on a space
X
. We shall say that d and d¢ are
(topologically) equivalent
metrics if every d-open set of
X
is d¢-open and every d¢-open
set of
X
is d-open.