Introduction to Topology
July 2024
Problem Set VII
1. A collection A of subsets of N has the strong finite intersection property (SFIP) if, and only if, for every finite subset A0 ⊆ A , we have that ∩ A0 is infinite.
Prove that if A ⊆ P(N) has the SFIP, then it is included in anon-principal ultrafilter. Hint Zorn’s Lemma.
2. Let X be a topological space and let F be a filter on N.
We will say that the sequence ~x = ⟨xn : n ∈ N⟩ F-converges to x, denoted by
xn → F x or x = lim xn ;
n→F
if, and only if, for every open neighborhood U of x, {n ∈ N : xn ∈ U} ∈ F. We will denote by LimF (~x) := {x ∈ X : xn → F x} .
(a) Prove that xn → x if, and only if, xn → Fr x, where Fr is the Frechet filter.
(b) Prove that~ X is a compact space if,and only if, for every ultrafilter U on N, and every sequence x = ⟨xn : n ∈ N⟩ of elements of X , the set LimU (~x) is non-empty.
(c) Prove that X is a Hausdorff space if, and only if, for every ultrafilter U on N, and every sequence ~x = ⟨xn : n ∈ N⟩ of elements of X , the set LimU (~x) has at most one point.
(d) Conclude that if X is a compact Hausdorff space then for every ultrafilter U on N, and every sequence ~x = ⟨xn : n ∈ N⟩ of elements of X , the set LimU (~x) has a unique point. This point is called the U -limit of the sequence ~x.
3. Prove that if X is a compact subspace of the Sorgenfrey line, then X is countable.
4. (a) Prove that if X is a compact Hausdorff space, x ∈ X , and U is an open neighbourhood of x, then there exists some open neighborhood V of x such that V ⊆ U.
Recall that a subset A of the topological space X is dense if, and only if, A = X .
(b) Prove that if X is a compact Hausdorff space and {Un : n ∈ N} is a countable collection of dense open subsets of X , then their intersection
∩ Un n∈N
is dense in X .
5. Given n ∈ N and ~a ∈ {0; 1}n , let
[~a] := {~x ∈ {0; 1}N : ⟨x1 ;:::;xn ⟩ = ~a}:
(a) Prove that the collection
{[a] : a ∈ {0, 1}n for some n ∈ N}
is a basis for the product topology on {0, 1}N .
A topological space is zero-dimensional if, and only if, it has a basis consisting of clopen sets.
(b) Prove that {0, 1}N is zero-dimensional. Hint Prove that each [a] is also closed.
(c) Prove that a zero-dimensional space with closed points is totally disconnected.
(d) Prove that {0, 1}N is compact.
Hint If O is an open cover of {0, 1}N without a finite subcover, recursively construct a sequence x ∈ {0, 1}N such that for all n ∈ N, the basic open neighbourhood [⟨x1,..., xn⟩] of ⃗x cannot be covered by finitely many elements of O; acontradiction.
(e) Prove that {0, 1}N has no isolated points.
(f) Conclude that {0, 1}N is uncountable.
6. Let X and Y be topological spaces with Y compact Hausdorff:
(a) Prove that the projection π 1 : X × Y → X is aclosed function.
(b) Prove that a function f : X → Y is continuous if, and only if, its graph
Γf := {⟨x,f(x)⟩ : x ∈ X}
is a closed subset of X × Y.