** 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 \( \rho(x,A)= \inf \{
\rho(x,a)\colon x \in A\} \).