Download TOPOLOGY: PROBLEM SET 7 Urysohn Metrization Theory

TOPOLOGY: PROBLEM SET 7
5th May, 2016
Urysohn Metrization Theory
Theorem (Urysohn metrization theorem). Every regular space X with countable basis is
metrizable.
Note. We will prove this theorem in several steps as we did with Urysohn Lemma. For
Exercises 1–3, we will assume that X is a second-countable regular topological space.
Exercise 1. Prove that there exists a countable collection of continuous functions fn : X →
[0, 1] having the property that, given any point x0 of X and any neighborhood U of x0 , there
exists an index n such that fn is positive at x0 and vanishes outside of U .
Exercise 2. Given the functions fn found in Exercise 1, take RN in the product topology
and defined a map F : X → RN by
F (x) := (f0 (x), f1 (x), f2 (x), . . .) .
Prove that F is a continuous, one-to-one function.
Date: 5th May, 2016.
Exercise 3. Let d (a, b) = min {|b − a| , 1} denote a metric on R. Let x = {xn }n∈N and
y = {yn }n∈N be two elements of RN and define
d (xi , yi )
.
D(x, y) := sup
i+1
Prove that D is a metric that induces the product topology on RN .
Exercise 4. Provide an example of showing that a Hausdorff space with a countable basis
need not be metrizable.
Exercise 5. Let X be a compact Hausdorff space. Show that X is metrizable iff X is
second-countable.
Hausdorff–Alexandroff Theorem
Exercise 6. Prove that every compact metric space is second-countable.
Exercise 7. Prove that if (X, d) is a compact metric space, then X is homeomorphic to a
subset of IN . Hint: Use ideas similar to your proof of the Urysohn metrization theorem.
Exercise 8. Prove that the Hilbert cube IN is the continuous image of the Cantor set.
Exercise 9. If K is a closed subset of the Cantor set, then K is the continuous image of
the Cantor set.
Exercise 10 (Hausdorff–Alexandroff Theorem). Prove that every compact metric space is
the continuous image of the Cantor set.
Exercise 11. Let δ > 0. Prove that there is a collection {Ci : i = 1, 2, . . . , k}, such that
each Ci isSa subspace of the Cantor space C that is homeomorphic to C, each diam Ci < δ,
and C = {Ci : i = 1, 2, . . . , k}.
Topological Groups
Definition. A topological group (G, τ ) is a group (G, ·, e) with a T1 topology such that the
map
m : (a, b) ∈ G×G 7→ a · b ∈ G
is continuous with respect to the product topology, and the map
i : a ∈ G 7→ a−1 ∈ G
is continuous.
Exercise 12. Give an example of a topological group and prove that your example is a
topological group.
Exercise 13. Prove that the topological closure of a subgroup is a subgroup. Give an
example of a subgroup of a topological group that is not topologically closed.
Exercise 14. Let G be a topological group. Prove that every open subgroup of G is also
closed.
Exercise 15. Let G be a topological group. Prove that if G is T1 , then G is Hausdorff.