Download TOPOLOGY PROBLEMS FEBRUARY 27, 2017—WEEK 2 1

TOPOLOGY PROBLEMS
FEBRUARY 27, 2017—WEEK 2
1. Topological spaces
1. Consider the set {1, 2, 3} with τ = {∅, {1}, {1, 2}, {3}, {1, 2, 3}}.
Can you add a single set to τ to make it a topology?
2. Define a topology on R as follows:
τ = {A ⊆ R | R \ A a finite set} ∪ {∅}
Show that (R, τ ) is a topological space.
3. Define a topology on R as follows:
τ = {(a, ∞) | a ∈ R ∪ {±∞}}
where we take (∞, ∞) to be ∅.
(i) Show that (R, τ ) is a topological space.
(ii) What is the closure of the set {0}?
(iii) Does the space have a base B ( τ ?
4. Consider the integers Z, and say that a set A ⊆ Z is open if it is
empty or if for every a ∈ A, there is some nonzero b ∈ N so that the
arithmetic sequence
S(a, b) = {a + bn | n ∈ Z}
is contained in A.
(i) Show that (Z, τ ) is a topological space.
(ii) Show that the sets S(a, b) form a base for the topology, and that
they are clopen.
(iii) Show that the complement of a finite set cannot be closed.
(iv) Prove that there are infinitely many primes.
2. Continuous functions, subspaces
5. Suppose f : X → Y is a continuous map. Show that the inverse
image of any closed set in Y is closed in X.
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6. Show that a map f : X → Y between topological spaces X and Y
is continuous if and only if for every subset A ⊆ X, f [Ā] ⊆ f [A].
7. Show that if g : X → Y and f : Y → Z are continuous, then
f ◦ g : X → Z is continuous.
8. Consider the integers Z, and the rational numbers Q, as subsets of
R, where the latter has the usual metric topology.
(i) What is the closure of Q in R?
(iii) Show that for any finite subset X of R, any point in X is clopen
in the subspace topology.
(ii) Show that every point in Z is clopen in the subspace topology. Is
the same true for any countable subset of R?
9. Let X be a topological space, and A a subset of X with the subspace topology. Let i : A → X be the inclusion map. Show that a map
f : Y → A is continuous if and only if i ◦ f : Y → X is continuous.
10. Suppose that X is a topological space with base B, and Y is a
subset of X. Show that the set
BY = {B ∩ Y | B ∈ B}
is a base for the subspace topology on Y .