431/531 - Class Outlines, Homework, and Handouts

Math 431/531 - Class Outlines, Homework, and Handouts

For further information on the class including Prof. Sinha's contact information, please read the syllabus. All references to a text below, for example by section numbers, refer to Topology by Munkres unless otherwise noted.

Our final exam will be Thursday December 10 at 10:15 am, in our normal classroom.
To help you prepare, here are some review problems. and solutions to TF questions.
We will discuss these review problems, some things I'd like to clarify from the homeworks and any questions you have on Tuesday at 6pm in our classroom.

Week Ten, Nov 30- Dec 4:

On Monday you'll finish your worksheets and then discuss Lebesgue number.
On Wednesday, we'll establish the contraction mapping theorem and start using it to define fractals in the plane.
On Friday we'll finish developing the mathematics underlying some fractals and then do a worksheet finding collections of maps which define fractals.

Last homework will be due on the last day of classes.

Week One, Sept 27-Oct 1:

Starting a class in topology is difficult because if one starts with an abstract definition, it seems unmotivated without examples, but it is hard to manage examples without a number of precise definitions and theorems. Hence, our first weeks will be a juggling act.

No classes on Monday.
On Wednesday we will start with some fun theorems from topology which you can share with your friends at parties. It will be quite a while before we can prove such theorems, but it is nice to know that the hard work we put in to formalism ultimately supports some remarkable, surprising results about familiar objects. We will also review some theorems which are not easy to share at parties, but which are some of the biggest theorems which use or involve topology. We will then move on to the basic definition of a topological space - in section 12 - and some examples.
On Friday we will look at topologies on finite sets, the finite complement topology, the Zariski topology, and the standard topology on R², which is an example we will use extensively.

Here is the first homework, which is due in class on Friday October 9th.

Week Two, Oct 5-Oct 9:

Defining topological spaces by listing all of their open sets is cumbersome. On Monday we will introduce two important ways of defining topological spaces, namely through bases and through the subspace topology. Our treatment will be fast, requiring you to go back over lecture on your own to fill things in afterwards if you want to follow properly.
We will spend more time on bases and on the subspace topology. We will also introduce the product topology through worksheets in the last twenty minutes of class.
We will finish the worksheets on the product topology, go over some of the problems as a class and then discuss closed sets. Reminder: first homework is due.

Week Three, Oct 12-Oct 16:

First graduate-student-only homework is due on Friday.

On Monday we will develop closed sets, the example of the Cantor set, and the notions of interior and closure.
Wednesday: we will discuss closures, interiors, and boundary points and start on a worksheet on them.
We will finish the worksheet and then start discussing continuity.

Week Four, Oct 19-Oct 23:

Second homework is due on Friday. Some solutions.

On Monday we will introduce continuity.
Wednesday: we will further develop continuity, giving a number of basic ways to construct new continuous functions out of others and verify a given function is continuous.
Friday: we will introduce metric spaces, giving a wide variety of examples

Week Five, Oct 26-30:

Contrary to what was announced, there is no grad student homework this week since our first mid-term is next Monday. Instead, all students might find it helpful to work on the mid-term review problems.

We develop the notion of convergent sequence and show that in a metric space, "sequential continuity" is equivalent to continuity.
On Wednesday, we'll talk about different notions of "infinite-dimensional Euclidean space." Some are easy to understand, coming from metrics which immediately extend familiar ones. Some are harder to understand but still metrizable. And some aren't even metrizable!

Week Six, Nov 2-6:

Third homework is due on Monday Nov 9, with Friday's after-class office hours moved to that day as well. Solutions.

Midterm exam is on Monday.
On Wednesday, we introduce the quotient topology, the origins of the notion that topology is "cut and paste geometry."
On Friday, more on the quotient topology, including a worksheet.

Week Seven, Nov 9-13:

No graduate student homework due this week.

On Monday we'll do some detailed, elementary proofs involving the quotient topology.
On Wednesday, we'll start establishing the classification of surfaces, working from the following paper by Dick Koch.
On Friday, we'll hopefully finish the classification of surfaces.

Week Eight, Nov 16-20:

Homework due Monday the 23rd. And solutions.

On Monday we'll resume "point-set" topology, discussing connectedness. Professor Botvinnik will be lecturing.
On Wednesday, we'll prove that the real numbers are connected.
On Friday we'll finish our discussion of connectedness, developing notions such as components and local connectedness.

Week Nine, Nov 23-25:

On Monday we'll start discussing compactness, one of the most useful properties of a topological space. We'll relate it to being closed and bounded.
On Wednesday, we'll show that a number of important spaces, starting with the interval, are compact. We'll also continue to relate topological compactness with notions coming from metric spaces.
No class on Friday - enjoy the holiday!