Math 5620 Spring 2018

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Course Information: Syllabus

Handout from last semester on ordinals. Notes on metric spaces.

Homework 1. (Due Thursday, 2/1/2018)
1.) Let $$X=\{ (x,y) \in {\mathbb{R}}^2 \colon x^2+y^2 \leq 1\}$$, with the subspace topology from $${\mathbb{R}}^2_{\text{std}}$$. Let $$\sim$$ be the equivalence relation on $$X$$ whose only non-trivial class is $$\{ (x,y)\colon x^2+y^2=1\}$$. Show that the quotient space $$X/\sim$$ is homeomorphic to $$S^2=\{ (x,y,z) \colon x^2+y^2+z^2=1\}$$ as a subspace of $${\mathbb{R}}^3_{\text{std}}$$.

2.) Let $$\sim$$ be the equivalence relation on $${\mathbb{R}}_{\text{std}}$$ with one non-trivial class which is equal to $$\mathbb{Z}$$. Show that $${\mathbb{R}}_{\text{std}}/\sim$$ is not first countable. Thus, there is a closed, continuous image of a second countable metric space which is not first countable.

3.) Suppose $$X_\alpha$$ for $$\alpha \in \mathcal{I}$$ are $$T_2$$ spaces with at least two points. Show that if $$| \mathcal{I} | >c$$ then $$\prod_{\alpha \in \mathcal{I}} X_\alpha$$ is not separable. [Hint: Construct pairwise disjoint opensets $$U_\alpha, V_\alpha$$ for $$\alpha \in \mathcal{I}$$, and consider the map $$\alpha \mapsto D\cap V_\alpha$$, where $$D$$ is a dense set in $$X=\prod_\alpha X_\alpha$$.]

Homework 2.
1.) Prove the following "easy version" of Urysohn's lemma for metric spaces: Let $$X,\rho$$ be a metric space, and $$A, B\subseteq X$$ be disjoint closed sets. Let $$f(x)= \frac{\rho(x,A)}{\rho(x,A)+\rho(x,B)}$$. Show that $$f$$ is a well-defined continuous function from $$X$$ to $$[0,1]_{\text{std}}$$, and that $$f(A)=0$$ and $$f(B)=1$$. Recall that $$\rho(x,A)= \inf \{ \rho(x,a)\colon x \in A\}$$.

2.) Show that the Tychnoff plank $$T$$ is an increasing countable union $$T=\bigcup_n X_n$$ of closed subspaces, each of which is $$T_4$$ in the subspace topology. Thus, an increasing union of closed $$T_4$$ subspaces need not be $$T_4$$.

3.) Text, exercise 15C, part (1).

First test will be on Thursday, March 22. Review sheet for first test: review.

Homework 3.
1.) Let $$X$$ be a set, and $$\mathcal{F}_1$$, $$\mathcal{F}_2$$ filters on $$X$$. Show that if $$\mathcal{F}_1 \cup \mathcal{F}_2$$ is an ultrafilter on $$X$$, then either $$\mathcal{F}_1$$ or $$\mathcal{F}_2$$ is an ultrafilter on $$X$$.

2.) a.) Show that if $$X,\tau$$ is a compact $$T_2$$ space, then there is no strictly finer topology $$\tau'$$ on $$X$$ for which $$X$$ is compact.

b.) Give an example of two $$T_1$$ topologies $$\tau \subseteq \tau'$$ on a set $$X$$ such that $$X$$ is compact under both of these topologoes and $$\tau \neq \tau'$$.

3.) Show that every compact subset of $$E$$ is countable. [hint: Show that if $$A\subseteq \mathbb{R}$$ is uncountable, then there is an $$x \in \mathbb{R}$$ which is a limit point of $$A$$ from the left, that is, $$\forall \epsilon >0 \, A\cap (x-\epsilon,x)\neq \emptyset$$.]

Homework 4.
1.) Show that that the Sorgenfrey line $$E$$ is Lindelof. [hint: Let $$\{U_\alpha \}_{\alpha \in \mathcal{I}}$$ be an open cover of $$E$$. It suffices to show that $$[0,1]$$ has a countable subcover. Let $$A=\{ x \in [0,1] \colon [0,x]$$ has a countable subcover $$\}$$. Let $$a = \sup(A)$$. Show that $$a \in A$$, and consequently $$a=1$$.]

2.) The Alexander subbase theorem say that a space $$X$$ is compact iff every cover of $$X$$ by subbasic open sets has a finite subcover. Prove this theorem (assuming AC) by following the following outline. Let $$\mathcal{S}$$ be a subbasis for $$X$$. Let $$\mathcal{B}$$ be the collection of all finite intersections of sets from $$\mathcal{S}$$, so $$\mathcal{B}$$ is a base for $$X$$. It suffices to show that every cover of $$X$$ by basic open sets has a finite subcover, so let $$\{ B_\alpha \}_{\alpha \in \mathcal{I}}$$ be a cover of $$X$$ by basic open sets.
By AC we may assume $$\mathcal{I} =\theta$$ is an ordinal, so our basic cover is of the form $$\{ B_\alpha \}_{\alpha <\theta}$$. Towards a contradiction, assume there is no finite subcover. By AC again, for each $$\alpha <\theta$$ let $$B_\alpha=S^\alpha_1\cap \cdots \cap S^\alpha_{k(\alpha)}$$ where the $$S^\alpha_j$$ are subbasic open sets in $$\mathcal{S}$$. By induction on $$\alpha <\theta$$ define integers $$k(\alpha) \in \{ 1, \dots, n(\alpha)\}$$, maintaining the following inductive hypothesis $$P(\delta)$$:

1.) The $$k(\alpha)$$ have been defined for all $$\alpha <\delta$$.
2.) The cover $$\{ S^\alpha_{k(\alpha)} \}_{\alpha <\delta} \cup \{ B_\alpha\}_{\alpha \geq \delta}$$ has no finite subcover.

Establish $$P(\delta)$$ for $$\delta \leq \theta$$ by induction on $$\delta$$, and show that $$P(\theta)$$ gives a contradiction.

3.) Consider the space $$E\times E$$. Let $$L=\{ (x,-x) \colon x \in \mathbb{R} \} \subseteq E\times E$$. Let $$A=\{ (x,-x)\colon x \in \mathbb{Q} \}$$, and $$B= \{ (x,-x)\colon x \in \mathbb{R}-\mathbb{Q} \}$$. Show that $$A$$, $$B$$ are disjoint closed sets in $$E\times E$$ which cannot be separated by disjoint open sets. This gives another proof that $$E \times E$$ is not $$T_4$$.