Math 4500/5600 Spring 2017

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Course Information: Syllabus

This is a one semester course in topology. We will be using Topology, a first course by James Munkres as the official text, but we will also be supplementing this with extra material covered in class. Homework problems will both come from the text as well as be assigned in class. In both cases, the problems will be listed here.

In the problems below you can use the following axioms for \( \mathbb{R}\).

Order axioms
\( (\mathbb{R},<) \) is a complete linear order with no smallest or largest element. Also, the set \( \mathbb{Q} \subseteq \mathbb{R}\) is dense in \( \mathbb{R}\) (that is, if \( x < y \) are reals, then there is a \( q \in \mathbb{Q}\) with \( x < q < y \) ).

Algebraic axioms
\( (\mathbb{R},+,\cdot ) \) is a field. That is, \( +, \cdot\) satisfy the usual algebraic laws:
1.) \( +\) and \( \cdot \) are commutative and associative.
2.) We have the distributive law \( x\cdot (y+z)=x \cdot y + x \cdot z\).
3.) There is an additive identity \( 0 \) and a multiplicitive identity \( 1\), and \( 0 \neq 1\).
4.) Every \( x \in \mathbb{R}\) has an additive inverse \( -x\), and every non-zero \( x \in \mathbb{R}\) has a multiplicitive inverse \(\frac{1}{x}\).

\( (\mathbb{R},+,\cdot ) \) is also an ordered field. That is, is we let \( P=\{ x \in \mathbb{R} \colon x>0 \}\) be the set of positive elements, then
1.) The sum or product of two positive elements is positive.
2.) If we let \( N= \{ -x \colon x \in P\} \), then \( \mathbb{R} \) is the disjoint union of \( N\), \( \{ 0\} \), and \( P\).
3.) \( x < y \) iff \(y-x \in P\).

Homework 1
1.) Show that if \( x \in \mathbb{R}\) then there is an \( n \in \mathbb{N}\) with \( x < n\) [hint: assume not, then \( \mathbb{N}\) has an upper bound in \(\mathbb{R}\). Get a contradiction from this.]

2.) Show that if \( x >0\) then there is an \( n \in \mathbb{N} \) such that \( 0 < \frac{1}{n} < x\) (the Archimedian property). [hint: use the first exercise. You will also need to show that if \( 0 < a< b\) then \( \frac{1}{a} > \frac{1}{b} \).]

3.) Show that if \( A \subseteq \mathbb{R}\), then \( x\) is a limit point of \(A\) iff there is a sequence \( \{ a_n\}_{n \in \mathbb{N}} \) with each \( a_n \in A\) and \( a_n \neq x\) such that \( a_n \to x\) (that is, the sequence converges to the point \( x\) ).

4.) Show that if \( F\subseteq \mathbb{R}\) is a closed set and \(F\) is bounded above, then \( \sup(F)\in F\).


Homework 2 Due Thursday, March 2.
1.) Let \( X= {\mathbb{R}}^2\). Define \( \rho ((x_1,y_1),(x_2,y_2))= \begin{cases} 1 & \text{if } x_1 \neq x_2 \\ 1 & \text{if } x_1=x_2 \wedge |y_1-y_2|\geq 1\\ |y_1-y_2| & \text{if } x_1=x_2 \wedge |y_1-y_2|<1 \end{cases} \). Show that \( \rho\) is a metric on \( {\mathbb{R}}^2\).

2.) Let \( \rho\) be the metric on \( {\mathbb{R}}^2 \) from (1). Show that the \( \rho\) metric topology on \( {\mathbb{R}}^2 \) is the same as the order topology from the lexicographic ordering on \( {\mathbb{R}} \times {\mathbb{R}} \).
[This space was considered in an example in class.]

3.) Show that in the space \( E\) (the Sorgenfrey line), any collection of pairwise disjoint open sets must be countable.

4.) Exercise 1 on page 83 of the text.


First test on Thursday, March 9.

Take-home part due Tuesday, March 28 at class time.

Homework 3
1.) Consider the real line with the Sorgenfrey topology. Let \( \{ x_n\}\) be a sequence in the Sorgenfrey line (so, the \(x_n\) are real numbers), and \( x\) a real number. Give an \( \epsilon\), \(\delta\) type statement which is equivalent to saying that the sequence \( \{ x_n\}\) converges to \(x\) in \( E\).

2.) Say a set \( U\) in a topological space is regular open if \( U= \text{int}\ \text{cl} (U)\, \).
a.) Give an example of two regular open sets in \( \mathbb{R}_{\text{std}}\) whose union is not regular open.
b.) Show that the intersection of two regular open sets in a topological space is regular open.

3.) Exercise 19 (a) and (b) on page 102 of the text.


Homework 4
1.) Show that a function \( f \colon Z \to X_1 \times X_2 \) is continous iff each of the maps \( \pi_1\circ f\), \( \pi_2 \circ f\) are continuous. Here, \( Z\), \( X_1\), and \( X_2\) are topological spaces and \( \pi_1 \colon X_1 \times X_2 \to X_1\) is the projection map from the product to \( X_1\), and likewise for \( \pi_2\).

2.) a.) Let \( X \) and \( Y\) be topological spaces. If \( f \colon X \to Y\) is continuous, show that if whenever \( x_n \to x\) in \( X\), then \( f(x_n)\to f(x)\) in \( Y\).

b.) Suppose that \( X\) is first countable, \( f \colon X \to Y\), and whenever \( x_n \to x\) in \( X\), then \( f(x_n)\to f(x)\) in \( Y\). Show that \( f\) is continuous.

3.) Show that if \( f \colon X \to Y\) and \( Y \subseteq Z\) (with \( Y\) having the subspace topology from \( Z\) ) and \( f\) is continuous as a function from \( X\) to \(Z\), then \( f\) is continuous as a function from \( X\) to \(Y\). [note: we proved the other direction in class.]

Final Exam on Tuesday, May 9, 10:30-12:30