Math 5610 Fall 2018
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Course Information:
Syllabus
Handout on ordinals.
Homework 1. (Due Tuesday, September 11)
1.) Suppose \( \tau_1\) and \( \tau_2\) are topologies on a set \( X\).
Show that \( \tau_1 \subseteq \tau_2 \) iff \( \forall U \in \tau_1\
\forall x \in U\ \exists V\in \tau_2\ (x \in V \subseteq U) \). Try to avoid using
AC.
2.) Let \( X\) be a set and \( A\mapsto \overline{A} \) an operation on the subsets of
\( X\) satisfying:
\(
\begin{align}
&1.)\ A \subseteq \overline{A} \\
&2.)\ \overline{\overline{A}} =\overline{A}\\
&3.)\ \overline{A\cup B}=\overline{A}\cup \overline{B}\\
&4.)\ \overline{\emptyset}=\emptyset
\end{align}
\)
Let \( \tau\) be defined by \( A\in \tau\) iff \( \overline{X-A} =X-A\).
You may assume \( \tau\) is a topology on \( X\). Show that
\( \text{cl}_\tau(A)=\overline{A}\) for all \( A\subseteq X\).
3.) a.) Show for any toplogical space \( (X,\tau) \) and any set \( A\subseteq X\)
we have \( \text{int cl int cl}(A)= \text{int cl(A)}\).
b.) Let \( X=\mathbb{R}_{\text{std}}\), and let \( A= (\mathbb{Q} \cap (-2,-1)) \cup \{ 0\}
\cup (1,2) \cup (2,3)\). Show that starting with the set \( A\) and using the operations of
complement, interior, and closure, exactly \( 14\) distinct sets are produced (the maximum
possible).
Homework 2. (Due Tuesday, September 25)
1.) A set \( A\) in a toplogical space \( (X, \tau)\) is called nowhere dense
if \( \text{int cl}(A)=\emptyset\).
a.) Show that \( A\subseteq X \) is nowhere dense iff
for every \( U\in \tau\) with \( U \neq \emptyset\) there is a \( V\subseteq U \), \( V \neq \emptyset\),
with \( V\in \tau\) such that
\( V\cap A=\emptyset\).
b.) Show \( F\subseteq X\) is closed nowhere dense iff \( X-F\) is open dense.
2.) Let \( X,\tau\) be a topological space. If \( A\subseteq X\), we define the
boundary \( \partial A\) of \( A\) by: \( \partial A= \overline{A}
\cap \overline{X-A}\). Note that \( \partial A=\partial (X-A)\).
a.) Show that \( \partial A= \overline{A}- A^\circ\).
b.) Show that if \( A\) is open or closed then \( \partial A\) is a closed nowhere dense set.
c.) Show that if \( A\) is a closed nowhere dense set then \( \partial A=A\).
[hint: for parts b and c, note that for a closed set \( F\), \(\partial F=F-\text{int}(F)\)].
3.) Let \( \rho\) be a metric on a set \( X\). Define \( d(x,y)=
\frac{\rho(x,y)}{1+\rho(x,y)} \). Show that \( d\) is a metric on \( X\)
and gives the same topology on \( X\) as \( \rho\).
Homework 3. (Due Tuesday, October 9)
1.) Let \( X=\mathbb{R}\). For each \( x \in X\), define the collection
\( \mathcal{C}_x\) as follows: if \( x \in \mathbb{Q}\), then
\( \mathcal{C}_x= \{ (x-\epsilon, x+\epsilon)\cap \mathbb{Q} \colon \epsilon >0 \} \).
If \( x \in \mathbb{R}-\mathbb{Q}\), let
\( \mathcal{C}_x= \{ (x-\epsilon, x+\epsilon)\cap (\mathbb{R}-\mathbb{Q}) \colon \epsilon >0 \} \).
a.) Show that for each \( x \in \mathbb{R} \) the \( \mathcal{C}_x\) satisfy the generalized
neighborhood base condition, and thus the \( \mathcal{C}_x\) define a topology \( \tau \) on \( X\).
b.) Show that all of the sets \( C \in \mathcal{C}_x\) are open for all \( x\).
c.) Compute \( \text{cl}_\tau(\mathbb{Q}) \) and \( \text{cl}_\tau(\mathbb{R}-\mathbb{Q}) \).
2.) For the space \( (X,\tau) \) of problem (1), determine if the space is
separable, if it is first countable, and if it is second countable.
3.) Let \( (X,\tau)\) be a topological space and \( A \subseteq X\). The
simple extension \( \tau_A\) of \( \tau\) by \( A\) is
given by \( \tau_A= \{ U \cup (V \cap A) \colon U, V \in \tau \} \).
a.) Show that \( \tau_A\) is a topology.
b.) Show that \( \tau_A\) is the smallest topology containing \( \tau\) and \( A\).
c.) Show that if \( \mathcal{B}\) is a base for \( \tau\), then
\( \mathcal{B} \cup \{ B\cap A\colon B\in \mathcal{B} \} \) is a base for \( \tau_A\).
Homework 4. (Due Thursday, November 8)
1.) Show that if \( (X, <_X) \) and \( (Y, <_Y) \) are linear orders and \( \pi \colon X \to Y\)
is an order-isomorphism, then for any \( x \in X\) we have that \( \pi \restriction I_x \)
is an order-isomorphism from \( I_x\) to \( I_{\pi(x)}\). [Here \( I_x\) is the initial segment
determined by \( x\) in \( (X,<_X) \), and likewise \( I_{\pi(x)}\) is the intial segment determined by
\( \pi(x)\) in \( (Y,<_Y)\) ].
2.) Exercise 9 from the notes: show that if \( \pi \colon \alpha \to \beta \) is order-preserving
(here \( \alpha,\beta\) are ordinals) then \( \alpha \leq \beta \). [hint: Prove by induction on
\( \alpha \), so let \( \alpha\) be least such that there is a counterexample. So assume \(
\beta <\alpha\). Apply induction to \( \pi \restriction \beta\) to deduce
\( \text{ran}(\pi\restriction \beta) \) is unbounded in \( \beta\), and this gives a contradiction.]
3.) Show that if \( \{ U_n\}_{n \in \mathbb{N}} \) is a sequence of open sets in
\( [0, \omega_1) \) and \( \bigcup_n U_n=[0,\omega_1) \), then there is an \( n \in \mathbb{N}\)
such that \( U_n\) contails a tail of \( [0,\omega_1 ) \), that is,
\( \exists \alpha<\omega_1 \forall \beta>\alpha\ (\beta \in U_n) \).
[hint: construct an increasing sequence \( \alpha_i \) of countable ordinals such that
for each \( n\), \(\alpha_i \notin U_n\) for infinitely many \( i\), and then consider
\( \sup_i \alpha_i \).]
Homework 5. (Due Thursday, November 29)
1.) Show that \( f \colon E\to \mathbb{R}_{\text{std}} \) is continuous iff
for all \( x \in \mathbb{R}_{\text{std}} \), if \( ( x_n) \) converges to \( x\)
from the right ( so \( x_n \geq x\) for all \( n\) ) then \( f(x_n)\) converges to
\( f(x) \). (here "converges" and "converges from the right" have the usual
meanings in \( \mathbb{R}_{\text{std}} \) ).
2.) Let \( X\) and \( Y\) be topological spaces, and suppose \( \{ F_\alpha\}_{\alpha \in \mathcal{I}} \)
is a pairwise disjoint collection of closed sets in \( X\). Suppose for each \( \alpha \in \mathcal{I}\)
that \( f_\alpha \colon F_\alpha \to Y\) is continuous. Define \( f \colon
\bigcup_{\alpha \in \mathcal{I}} F_\alpha \to
Y\) by \( f(x)= f_\alpha(x) \) for \( x \in F_\alpha\).
a.) Show that \( f\) is continuous if the \( F_\alpha\) satisfy the following: for every
\( x \in \bigcup_{\alpha \in \mathcal{I}} F_\alpha \) there is a neighborhood \( U\) of
\( x\) in \(X\) which intersects only finitely many of the \( F_\alpha\).
b.) Give an example to show that in general \( f\) need not be continuous.
3.) Show that if \( f\colon X\to Y\) is an embedding, and \( g \colon Y \to Z\)
is an embedding, then \( g\circ f\colon X\to Z\) is an embedding.
Homework 6.
1.) Let \( X_\alpha\), for \( \alpha \in \mathcal{I} \), be topological spaces,
and let \( X=\prod_{\alpha \in \mathcal{I}} X_\alpha\). Let \( \alpha_0 \in \mathcal{I}\), and for \(
\alpha \neq \alpha_0\) let \( b_\alpha \in X_\alpha\) be fixed points. Let
\( \tilde{X}_{\alpha_0} \subseteq X\) be the subspace
\( \tilde{X}_{\alpha_0} =\{ x \in X \colon x(\alpha)= b_\alpha \ \forall \alpha \neq \alpha_0\} \).
Show that \( \tilde{X}_{\alpha_0} \) is homeomorphic to \( X_{\alpha_0}\).
2.) Let \( X\) be a topological space and let \( \{ f_\alpha \}_{\alpha \in \mathcal{I}}\)
be a family of functions \( f_\alpha \colon X \to \mathbb{R}_{\text{std}}\), and suppose
\( X\) has the weak topology from \( \mathcal{F}\). Let \( \mathcal{G}\) be the family of functions
of the form \( g= (f_{\alpha_1}-q_1)^2+\cdots+ (f_{\alpha_k}-q_k)^2 \) where \( q_i \in \mathbb{Q}\)
and \( \{ \alpha_1,\dots,\alpha_k\}\) is a finite subset of \( \mathcal{I}\).
Show that \( \mathcal{G}\) separates points from closed sets in \( X\).
3.) This exercise generalizes the fact that a map \( f\) into a product space is continuous iff each
coordinate function \( \pi_\alpha \circ f\) is continuous. Let \( (X, \tau)\) be a topological space,
let \( \{ (X_\alpha, \tau_\alpha)\}_{\alpha \in \mathcal{I}} \) be topological spaces, and let
\( f_\alpha \colon X\to X_\alpha\) be maps and assume \( X\) has the weak topology from the
\( f_\alpha\). If \( Y\) is a topological space, then show that a map \( f \colon Y\to X\)
is continuous iff each map \( f_\alpha \circ f \colon Y \to X_\alpha\) is continuous.