Math 5620 Spring 2010

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Course Information: Syllabus

Handout on ordinals.



Homework 1.) (due Thursday, Jan. 28, 2010).
1.) Show that subspaces and (arbitrary size) products of T_3¹⁄2 spaces are T_3¹⁄2.
2.) Text 14A parts 2, 3.
3.) Text 14D.



Homework 2.) (due Tuesday, Feb. 9, 2010).
1.) Let (X,τ) be a topological space. Show that the following are equivalent:
a.) There is a countable family F of continuous real-valued functions on X such that X has the weak topology from F.
b.) There is a countable family F' of continuous real-valued functions on X which separate points from closed sets.
Do not assume that X is T₁ in this problem.
2.) Let A ⊆ R_std be the set A={ ¹⁄_n : n ∈ N⁺}. Let τ be the simple extension of the standard topology making R-A open (i.e., τ is the smallest topology making A closed). Show that τ is second countable and T₂, but not metrizable (hint: show that τ is not T₃).
3.) Show that E (the Sorgenfrey line) is separable and T₄ but not metrizable (hint: if E were metrizable, then so would be E × E.
But consider the subspace of E × E given by { (x,-x): x ∈ R}.)

Homework 3.)
1.) Text 15B part 1.
2.) Text 15C part 1.
3.) Text 15E (hint: one direction is trivial. To show ( under the given assumption) A is C* embedded in X, follow the proof of the Tietze extension theorem but instead of using Urysohn's lemma, use the assumption).

Homework 4.)
1.) Text 11A parts 1,2,3.
2.) Text 11C.
3.) Let X= ∏_R R_std. Show that if A ⊆ X, and x ∈ cl_X(A), then there is a net { x _λ } _{λ ∈ Λ} from A which converges to x, where Λ is the directed set of all finite subsets of R ordered by inclusion.

Midterm Exam on Tuesday, March 9.

Homework 5.) (Due Thursday, March 25, 2010).
1.) Text 12B parts 1, 2.
2.) Text 12D parts 1, 2.

Homework 6.) (Due Tuesday, April 6, 1010)
1.) a.) Show that if X is countably compact then every infinite A ⊆ X has a limit point.
b.) Show that if X is T₁ and every infinite A ⊆ X has a limit point, then X is countably compact.
c.) Give an example of a T₀ space X which is not countably compact, but every infinite A ⊆ X has a limit point.
2.) Let a_n be a sequence of real numbers and consider the sequence of functions f_n(x)=sin(a_nx) (so each f_n is a function from R to R). Show that there is a function f: R → R which is limit of the f_n in the following sense: for every finite set of reals x₁,...,x_m and every ε>0, there are infinitely many n such that |f_n(y)-f(y)|<ε for all y ∈ {x₁,...,x_m}.
3.) Let (X,τ) be a compact T₂ space, and let τ ⊆ τ' be a finer topology. Show that if (X,τ') is also compact then τ=τ'.

Homework 7.)
1.) (Alexander Subbase Theorem): Let (X , τ) ba a topological space and let S be a subbase for the topology τ (i.e., the collection of finite intersections of sets from S is a base for τ). Show that (X, τ) is compact iff every cover of X by sets from S has a finite subcover. [hint: Suppose this property holds and let U={ U_α: α ∈ I} be an open cover of X. Without loss of generality (say why) we may assume each U_α is a basic open set, that is, a finite intersection of sets from S. By AC, we may assume I=θ is an ordinal. Assume U has no finite subcover. By induction on α ≤ θ show that there are S_β ∈ S for β < α such that { S _β : β < α} ∪ { U _β : β ≥ α} has no finite subcover. For α = θ this will be a contradiction].
2.) Use the Alexander Subbase theorem to give a direct proof of Tychnoff's theorem.
3.) Give a direct proof that a locally comapact T₂ space is T₃ (by direct we mean don't invoke the one-point compactification).

Homework 8.)
1.) Let (X, τ) be a T_3¹⁄2 space with a dense set D. Show that |β(X)| ≤ 2^2|D|.
2.) Let (X, τ) be a T₄ space which is not compact. Show that β(X) is not first countable. In fact show that for any x ∈ β(X)-X there is no countable base at x. [hint: Suuppose there were a countable base at x. Get a sequence { x_n } from X which converges to x. Let F={ x_n}. Note that F is discrete in X. Use the Tietze extension theorem to extend a suitable continuous function on F to X and then to β(X) to get a contradiction.