Math 5610 Fall 2009

(back to home)

Course Information: Syllabus

Handout on ordinals.



Homework 1.
1.) Let U be an open set in R_std. Show that U can be written as a countable, pairwise disjoint union of open intervals.

2.) Let X=(-1,1) ∪ (2,3) with the following topology. We define U ⊆ X to be in τ provided U is open in τ_std and if U ∩ (-1,1) ≠ ∅, then 0 ∈ U.
a.) Show that τ is a topology on X.
b.) Let A= { 0} ∪ (2,3). Compute A′.
c.) Show that A′ is not closed in X.
d.) Is τ T₀, is it T₁?

3.) Text problem 3A #2,4,5.

Homework 2.
1.) Text 3D, 1,2,3.
2.) Text 3G
3.) Let A^' denote the set of limit points of A, and define inductively A⁽ⁿ⁺¹⁾=(A⁽ⁿ⁾)^'.
a.) Give an example of a countable closed set F in R_std with the property that all F⁽ⁿ⁾ ≠ ∅ but ∩_n F⁽ⁿ⁾ = ∅.
b.) Give an example of a countable closed set F in R_std with the property that all F⁽ⁿ⁾ ≠ ∅ and also ∩_n F⁽ⁿ⁾ ≠ ∅.
Homework 3.
1.) Define ρ on R by: &rho(x,y)=&rho(x,y)+|x²-y²|.
a.) Show that ρ is a metric on R.
b.) Show that ρ and &rho_std give the same topology on R.
c.) Show that there is no K such that ρ ≤ K ρ_std.

2.) Text problem 2F (correct the statement in part 3).

3.) Show that the following three metrics on Rⁿ give the same topology. You don't have to show that they are metrics.
a.) ρ(x, y)= ((x₁-y₁)² + … + (x_n-y_n)²)^1/2.
b.) σ(x, y)=|x₁-y₁| + … + |x_n-y_n|
c.) d(x, y)= max(|x₁-y₁|,&hellip, |x_n-y_n|)

Homework 4.
1.) Let ρ be a metric on a set X.
a.) Show that d=ρ/(1+ρ) is also a metric on X.
b.) Show that d and ρ give the same topology on X.

2.) See the text 4C for the definition of the slotted plane.
a.) Show this definition satisfies the generalized neighborhood base criterior and thus gives a topology on R². Show that the basic sets are open in this topology.
b.) Is the slotted plane second countable? Prove your answer.

3.) a.) Show that the Moore plane is first countable.
b.) Show that the Moore plane is not second countable.

Homework 5.
Handout sheet on ordinals, exercises 1,2, 4.

Homework 6.
1.) Suppose S ⊆ On is a set of ordinals and sup(S) ∉ S. Show that sup(S) is a limit ordinal.
2.) Show that [0, ω₁] is neither first countable nor separable.
3.) A set C ⊆ ω₁ is said to be closed unbounded, or c.u.b., if C is closed (in the order topology) and unbounded, that is, C is not a subset of α for any α < ω₁.
Show that if { C_n } is a countanle collection of c.u.b. subsets of ω₁ then ∩_n C_n is also c.u.b. in ω₁.

Exam on Thursday, November 5.

Homework 7.
1.) Text 6A part 7.
2.) Text 6C.
3.) a.) Let (X,ρ), (Y.σ) be metric spaces. Show f: X → Y is continuous at the point x iff ∀ ε >0 ∃ δ>0 ∀ x' [0< ρ(x,x')<δ → σ (f(x),f(x'))< ε]

b.) Let X, Y be topological spaces and suppose B₁, B₂ are bases for X and Y respectively.
Show that a function f: X → Y is continuous at the point x iff for every basic open set V ∈ B₂ containing f(x), there is a basic open set U ∈ B₁ containing x such that f(U) ⊆ V.

4.) Characterize the continuous functions f: E &rarr R_std where E denote the Sorgenfrey line. Prove your answer carefully and completely.

Homework 8.
1.) Show that for every countable ordinal α, the space [0, α), with the usual subspace topology, is homeomorphic to a subspace of R_std.
2.) Show that [0, ω₁ ) is not homeomorphic to a subspace of R_std (hint: consider the image of the successor ordinals).
3.) Text 7K parts 1,4.