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