Math 5610 Fall 2017

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Course Information: Syllabus

Handout on ordinals.

Homework 1 (Due Tuesday, 9/26)
1.) a.) Show that if $$(X,\tau)$$ is $$T_1$$ and $$\tau$$ is closed under arbitrary intersections, then $$\tau$$ is the discrete topology on $$X$$.
b.) Let $$X=(0,1) \cup (2,3)$$ and let $$\tau$$ be the following collection: $$U \in \tau$$ iff $$[(U \cap (2,3) \neq \emptyset) \to ( (0,1) \subseteq U)]$$. Show that $$\tau$$ is $$T_0$$ but not $$T_1$$, and $$\tau$$ is closed under arbitrary intersections.

2.) Let $$X,\tau$$ be a topological space. Define a binary relation $$\sim$$ on $$X$$ by: $$x \sim y$$ iff $$\forall U\in \tau\ (x \in U \leftrightarrow y\in U)$$.
a.) Show that $$\sim$$ is an equivalence relation on $$X$$.
b.) Let $$Y=X/\sim$$ be the set of equivalence classes of $$X$$ with respect to $$\sim$$. Define $$\sigma=\{ V \subseteq Y\colon \cup V \in \tau \}$$. Here $$\cup V$$ denotes the union of the set $$V$$, that is the set of all elements of all the equivalence classes in $$V$$. Show that $$\sigma$$ is a topology on $$Y$$ and that it is $$T_0$$.

3.) Show that if $$(X,\tau)$$ is $$T_1$$ and $$A \subseteq X$$, then $$A'$$ is closed and $$A''\subseteq A'$$.

Notes on metric spaces: metric spaces, TeX file

Homework 2 (Due Thursday, 10/5)

First Test on Tuesday, October 17

Homework 3 (Due Tuesday, November 7).

1.) An open set $$U$$ in a topological space is said to be regular open if $$U=$$ int cl $$U$$.
a.) Show that for any set $$A$$ in a topological space we have that int cl int cl $$A$$ = int cl $$A$$. Show that the regular open sets are precisely the sets of the form int cl $$V$$ for some open set $$V$$.
b.) Show that the intersection of two regular open sets is regular open, but give an example of two regular open sets in a metric space whose union is not regular open.

2.) If $$(X,\tau)$$ is a topological space and $$A\subseteq X$$, then the simple extension of $$\tau$$ by $$A$$ is the topology $$\tau_A= \{ U \cup (V\cap A) \colon U,V\in \tau\}$$.
a.) Show that this is a topology, and is the smallest topology on $$X$$ containing $$\tau$$ and $$\{ A\}$$.
b.) Show that if $$(X,\tau)$$ is first countable then so is $$\tau_A$$.

3.) a.) Show that if $$<_1,\dots, <_n$$ are wellorderings on sets $$A_1,\dots,A_n$$, then lexicographic ordering $$<_{\text{lex}}$$ on $$A_1\times \cdots \times A_n$$ is a wellordering.
b.) Show that (a) fails for infinite products. Specifically, show that lexicographic ordering on $$2^{\mathbb{N}}$$ is not a wellordering.

Homework 4

1.) Suppose $$X$$ and $$Y$$ are topological spaces and $$\mathcal{B}, \mathcal{C}$$ are bases for $$X$$ and $$Y$$ respectively. Show that $$f \colon X\to Y$$ is continuous at a point $$x \in X$$ iff for every basic open set $$C \in \mathcal{C}$$ containing $$f(x)$$, there is a basic open set $$B\in \mathcal{B}$$ containing $$x$$ such that $$f(B)\subseteq C$$.

2.) Determine which functions $$f\colon \mathbb{R}_{\text{std}} \to E$$ are continuous. Prove that your answer is correct.

3.) Show that if $$f \colon [0,\omega_1 ) \to \mathbb{R}_{\text{std}}$$ is continuous, then $$\text{ran}(f)$$ is a bounded set in $$\mathbb{R}$$ (that is, $$\text{ran}(f)\subseteq [-N,N]$$ for some $$N$$ ).
[hint: if not, argue there is a least $$\alpha <\omega_1$$ such that $$\text{ran}(f \restriction \alpha)$$ is unbounded in $$\mathbb{R}$$, then get a contradiction from this.]

Homework 5

1.) a.) Show that if $$X=\prod_{\alpha \in \mathcal{I}} X_\alpha$$ is a product of topological spaces and $$F_\alpha \subseteq X_\alpha$$ is closed, then $$F=\prod_{\alpha \in \mathcal{I}} F_\alpha$$ is a closed set in $$X$$.
b.) Give an example to show that a product of open sets need not be open.

2.) Consider $$X=\prod_{\alpha \in \mathbb{R}} \mathbb{R}_{\text{std}}$$. Let $$A \subseteq X$$ be the set of convex functions. Recall $$f \colon \mathbb{R} \to \mathbb{R}$$ is convex if for all $$x< y$$ in $$\mathbb{R}$$ and all $$0 \leq p \leq 1$$ we have that $$f(px+(1-p)y) \leq pf(x)+(1-p)f(y)$$. Show that $$A$$ is a closed set in $$X$$.

3.) Suppose $$\mathcal{I} \subseteq \mathcal{J}$$ and for all $$\alpha \in \mathcal{J}$$ we have a topological space $$X_\alpha$$. Show that there is an embedding from the space $$X=\prod_{\alpha \in \mathcal{I}} X_\alpha$$ into the space $$Y= \prod_{\alpha \in \mathcal{J}} X_\alpha$$.

4.) Text, exercise 8I (parts 1 and 2). [For part 2 you can use a result proved in class.]