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Introduction to Topology
创建日期：2024-09-03 15:19:38
所属分类：数学mathematics
标签： Topology
Introduction to Topology

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Introduction to Topology

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.

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