Graphics Reference
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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.
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