Measure, Integration & Real Analysis, 2021a

Section 6A Metric Spaces 153
EXERCISES 6A
1 Verify that each of the claimed metrics in Example 6.2 is indeed a metric.
2 Prove that every finite subset of a metric space is closed.
3 Prove that every closed ball in a metric space is closed.
4 Suppose V is a metric space.
(a) Prove that the union of each collection of open subsets of V is an open
subset of V.
(b) Prove that the intersection of each finite collection of open subsets of V is
an open subset of V.
5 Suppose V is a metric space.
(a) Prove that the intersection of each collection of closed subsets of V is a
closed subset of V.
(b) Prove that the union of each finite collection of closed subsets of V is a
closed subset of V.
6 (a) Prove that if V is a metric space, f ∈ V, and r > 0, then B( f , r) ⊂ B( f , r).
(b) Give an example of a metric space V, f ∈ V, andr > 0 such that
B( f , r) ̸= B( f , r).
7 Show that a sequence in a metric space has at most one limit.
8 Prove 6.9.
9 Prove that each open subset of a metric space V is the union of some sequence
of closed subsets of V.
10 Prove or give a counterexample: If V is a metric space and U, W are subsets
of V, then U ∪ W = U ∪ W.
11 Prove or give a counterexample: If V is a metric space and U, W are subsets
of V, then U ∩ W = U ∩ W.
12 Suppose (U, d U ), (V, d V ), and (W, d W ) are metric spaces. Suppose also that
T : U → V and S : V → W are continuous functions.
(a) Using the definition of continuity, show that S ◦ T : U → W is continuous.
(b) Using the equivalence of 6.11(a) and 6.11(b), show that S ◦ T : U → W is
continuous.
(c) Using the equivalence of 6.11(a) and 6.11(c), show that S ◦ T : U → W is
continuous.
13 Prove the parts of 6.11 that were not proved in the text.
Measure, Integration & Real Analysis, by Sheldon Axler