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 \( T_4\).