Download List 4 - Math KSU

MATH 701 - ELEMENTARY TOPOLOGY I
Exercise list 4
1. Let S 1 = {(x, y) ∈ R2 x2 +y 2 = 1} be the circle, with the Euclidean metric induced from
R2 . Prove that there does not exist any proper subset A $ S 1 which is homeomorphic
to S 1 .
2. Let {Vk }k∈N be a collection of non-empty subsets of Rn satisfying, for all k ∈ N,
(a) Vk is closed and bounded; and
(b) Vk+1 ⊆ Vk .
Prove that ∩N Vk is a non-empty compact subset of Rn .
3. Let p : (X, TX ) → (Y, TY ) be a continuous, surjective map that sends closed subsets to
closed subsets. Suppose in addition that Y is compact and, for each y ∈ Y the subset
p−1 ({y}) is compact in X . Prove that X is compact (hint: if U is an open set containing
p−1 ({y}), check that there is an open subset W containing y and such that p−1 (W ) is
contained in U ).
4. Let (X, TX ) be a compact, metrizable space. Show that X is second-countable.
5. Let f : (X, TX ) → (Y, TY ) be a continuous map that sends open subsets to open subsets.
(a) Show that if X is first-countable (respectively second-countable) then f (X) is firstcountable (respectively second-countable).
(b) Give an example of a continuous function f : (X, TX ) → (Y, TY ) such that X is
first-countable but f (X) is not first-countable.
6. Let (X, TX ) be a topological space. Prove that ∆ = {(x, x) | x ∈ X} ⊆ X × X is closed
(in the product topology) if and only if X is Hausdorff.
7. Let X = R and let b = {[a, b) | a, b ∈ R, a < b}. This turns out to be a basis for a
topology on X , T . Let L = {(x, −x) | x ∈ X} ⊆ X × X . Prove the following.
(a) L is closed in the product topology.
(b) L has the discrete topology as a subspace of X × X .
8. Let f, g : (X, TX ) → (Y, TY ) be continuous functions, and suppose Y is Hausdorff. Show
that {x | f (x) = g(x)} ⊆ X is closed.
9. Let p : X → Y be a closed (i.e., p(A) ⊆ Y is closed for all A ⊆ X closed), continuous,
surjective map. Suppose in addition that p−1 ({y}) is compact for all y ∈ Y . Show the
following.
(a) If X is Hausdorff, so is Y .
(b) If Y is regular, so is Y .
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10. A space (X, TX ) is completely normal if every subspace is normal. Show that (X, TX ) is
completely normal if and only if the following holds:
• For all A, B ⊆ X satisfying ClA ∩ B = ∅ = A ∩ ClB , there exist U, V ∈ TX such that
A ⊆ U , B ⊆ V and U ∩ V = ∅.
Hint: if X is completely normal, consider (ClA ∩ ClB)c ⊆ X .
11. Let (X, TX ) be a compact Hausdorff space. Show that X is metrizable if and only if X
is second countable.
12. Let (X, TX ) be a compact Hausdorff space that is the union of two closed subsets,
X1 and X2 . Show that if X1 and X2 are metrizable, then so is X (hint: construct a
countable collection A of open sets of X whose intersection with Xi for a basis for Xi ,
for i = 1, 2; assume X1 \ X2 and X2 \ X1 belong to A; let b be the family of finite
intersections of elements of A.)
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