** 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)

Exercises 1,2,3 on page 4 of metric space notes (link to notes on this page).

**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.]