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Aut1997: MTG4302 3279 Elements.Topology.1 MWF6 LIT201
Aut1997: MTG5316 3280 Intro.Topology.1 MWF6 LIT201

Topology One Knotted Torus

This page has material for the 1st semester and, further below, for the 2nd semester course.


Our text is Topology (2nd edition).
Author: James Munkres ISBN: 978-0131816299
Year: 2000 Publisher: Prentice Hall
Photo of text cover

[Image: Coffee cup] 15 Dec 1997:    Here is the Inter-Semester HW Assignment.

Our topic for the second semester is: A gentle introduction to Algebraic Topology. Here is a list of the topics that students had to choose from:

[Image: Steaming coffee cup] Exams

While in-class exams are closed-book, take-home exams are open book/notes.

Hand-in by sliding your exam u n d e r room 402 Little Hall my office door (Little Hall 402, Northeast corner) . The take-home exams must be typed; if need be, mathematical symbols can be hand-written. Diagrams may be drawn by hand.

ExamText When Class-A (pdf): 27Oct1997 In-class. Metric spaces, Product topology. Home-B (pdf): 25Nov1997 Due: 8Dec1997. Class-C (pdf): 15Dec1997 In-class. (Monday night)

In a previous incarnation of this course (the old syllabus (pdf) is available), the students found the following documents useful.



Administrative Info

Kathy Reiff is maintaining the H-problem (txt) list, whereas I maintain the syllabus (pdf).

Suzanne Sheridan (with the able assistance of Ken Park) is maintaining the list of eddresses, addresses and telephone numbers.

Sean Bailey is in charge of the List of [Image: Coffee cup] Readings and HW assignments

Thao Tran keeps the extra copies of handout sheets. If you missed a class with a handout sheet, please get a copy of the sheet from her.

Todd Durham is maintaining the Solutions Notebook, which is currently in the magazine rack of the Resource Room. You can look at solutions while in this room, or you can check out the NB for upto two hours, so as to photocopy the sections that you want. Make sure to leave a (full) sheet of paper in the rack with your Name and the Date/Time that you took the NB.

Some Solutions

I've typeset several solutions to problems in the notes. In addition, below are solutions to some more advanced problems (not in our notes). These are listed by category: Problem numbers refer to the notes.

Metric space results

The notes construct a completion of a metric space; here we show that this construction indeed produces a space which is complete (pdf).

If a generalized Cantor set is metrizable, then it is homeomorphic to the standard Cantor set (pdf) Pic of Cantor Set.

The question What kinds of real-valued functions have a local extremum at every point? (pdf) has a partial answer.

Here is a problem: Can you construct a complete metric on the irrationals, equivalent to the usual metric? Indeed, one can characterize the completable subsets of a metric space (pdf); but don't look at this until you've seriously tried the above problem.

Compactness

One tool used in topological dynamics is the Ellis enveloping semigroup (ps). Of possible interest to the beginning student: The section on nets, pages 1-3 and the section on the Stone-Cech compactification, pages 11,12. (Unless you already know some topological dynamics, the middle section on the enveloping semigroup will seem unmotivated.)

I will pass out Paul Chernoff's short proof of the Tychonoff theorem, which uses nets. In the case of a countable product of finite discrete spaces, there is a simple proof using Cantor diagonalization. Some time ago I produced the World's Longest Proof of Tychonoff's Theorem (ps) by mimicking Cantor diagonalization for a general product. Perhaps you can compactify this proof?

The first third of a page of World's Longest… gives an example showing that sequential-compactness is not preserved under uncountable products (ps).

An example of a metric-space compactness argument is this simple condition for separability of the space of continuous functions (pdf) between two metric spaces.

Connectedness

There is an example where the connected component of a point is smaller than its pseudo-component (pdf).

There is a standard example of a denumerable Hausdorff space which is connected (pdf).

A rather surprising example was found by Cantor. There is a topological space (a cleverly chosen subset of the plane) which has an explosion point (ps). This means that the set explodes when this special point is removed: The space is connected -but, the removal of this special point renders the set totally-disconnected! (The construction here is the standard one; I've taken pains to not skip any steps in the proof —comments are most welcome.)

Miscellaneous results

Here is a classic Baire Category problem, that an eventually derivative-zero f must be a polynomial (ps).

Any compact manifold can be embedded in a finite-dimensional Euclidean space (pdf).

Student solutions Student holding HW

Below, are some of the solutions submitted by former students.



Spr1998: MTG4303 5045 Elem.Topology.2 MWF6 LIT203
Spr1998: MTG5317 6171 Intro.Topology.2 MWF6 LIT203

Topology Two Four linked rings

In the second semester, we are continuing with Munkres text, with a focus on Algebraic Topology. (Time permitting, we will do some knot theory too.) The syllabus, as well as additional exams and many solutions are on the Topology One page.

[Image: Steaming coffee cup] Exams

These are all take-home (open book/notes) exams. They are to be slid u n d e r room 402 Little Hall my office door (Little Hall 402, Northeast corner) office door. The exams must be typed. Diagrams may be drawn by hand. Here are Home-D (pdf) and Home-E (pdf):

[Image: Coffee cup] Homework thus far.

Jeremy Smith is keeping track of the new H-problems (html).

NOTE: Munkres uses a hyphen in naming a section of his text, e.g, section 3-6. However, I will use a point, i.e section 3.6, so that I can refer to a range of sections, e.g, read sections 3.6-3.8. In an assignment, ellipses … means that there is more forthcoming.

NoClass 19Jan, MLK Day. Due 21Jan98, Wednesday. We will meet in room LIT368 Review section 3.5. at 12:20PM. READ sections 3.6-3.8 DO: P.177: 1,4-6. Due 26Jan98, Monday. DO: P.177: 3. Due 02Feb98, Monday. Review 3-8,3-1. READ 3-2. DO: P.181: 1,2,3. Due 09Feb98, Monday. DO: P.186: 2,3,4. Also, H-108. Due 09Mar02, Monday. Do: P.158 5,9. P.163 2. Due 11Mar02, Wednesday. Do: P.163 10,11. (Optional hand-in.) Due 25Mar02, Wednesday. Do: P.215 3,7. P.220 5. Due 06Apr1998, Monday. Do: P.274 1. P.283 5,8,10. (Drill: 6,7. Sugg: 9) H-111 Due 20Apr1998, Monday. Review Pasting Lemma, P.108. Do: P.110 10. (Drill 9.) H-116.

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____End: Topology One   Page