** 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 \( T_4\).