MATH 420, Special Topics: Topology
Spring 2009
January 14, 2009
Bryan Smith
Contents
1 Introduction
2 Course Information
2.1 Textbook
2.2 Basic Information
2.2.1 Logistics
2.3 Examinations
2.4 Final Examination
2.5 Homework
2.6 Reading
2.7 Course Information Updates
2.8 Total Points
2.9 Ungraded Homework Assignment
1 Introduction
We will be covering the basics of "Pointset Topology" in
this course. Our material will focus on what type of
structures need to be placed on a set in order to make it a
"topological space" and what properties we can deduce from
these structures. You have already experienced a similar
situation in linear algebra where the structure of a
"vector space" (addition, scalar multiplication and the
definition with 10 properties) is placed on a set. Similar
things are done in Math 433 and 434 in defining groups,
rings, fields, modules, etc.
It is fair to say that pointset topology is a fundamental
part of mathematics in that almost every other branch makes
use of it at some level. The disciplines you are most
familiar with are interested in what topology can tell them
about "continuity" and related properties of functions,
For example, Physics, Statistics and Analysis require the
ideas associated with derivatives (which is more stringent
than continuity), modern Geometry requires continuity (and
differentiability) in its concepts of transformations,
curvature, and metrics, and almost every type of
optimization uses tools that rely on small changes in inputs
not having catastrophic effects on outputs of functions
(i.e., continuity). Hence, at an intuitive level, we will be
studying what structures are needed to make functions
between sets be "continuous" and
what other properties follow from those structures.
2 Course Information
2.1 Textbook
The textbook is Aspects of Topology, 2nd Ed,
Christenson and Voxman, ©1998, BCS Associates.
I expect that you already feel comfortable with the basic approaches to
doing a mathematical proof. However, if you would like to have a
resource at hand, you might consider one of the many books on
"how to do proofs" such as The Nuts and Bolts of Proofs.
2.2 Basic Information
You can find information pertinent to all of my classes at the link
below and, once there, information specific to this class by
clicking on the Math 420 link.
http://math.ups.edu/~bryans/ [1]
Professor Bryan Smith | Thompson 390D | 879-3562 | bryans[at]ups.edu |
|
Math 420A | Thompson 310 | Mon & Fri | 11:00 - 11:50 A.M. |
| Thompson 395 | Tue & Thu | 11:00 - 11:50 A.M. |
|
Office Hours | | Tue. | 3:00 - 3:50 P.M |
| | Wed. | 3:00 - 4:30 P.M. |
| | Thu. | 1:00 - 1:50 P.M. |
If you have trouble meeting during office hours please make an
appointment for a better time.
2.3 Examinations
Since the homework will be used to facilitate
deep understanding, tests will be used more as a basic check of your
knowledge. Hence, there will be three, "straightforward", semester
examinations that will include definitions as well as proofs.
Examination One | Thursday February 12 |
Examination Two | Thursday March 12 |
Examination Three | Thursday April 16
|
Study Session If there is enough interest we can
schedule an evening study session for the Wednesday
before an examination.
2.4 Final Examination
12:00 - 2:00 P.M. | Monday | May 11
|
The final will also be "straightforward" and comprehensive. The
final cannot be rescheduled so do not plan plane flights (or
anything else) that will conflict with it.
2.5 Homework
Each of you will be assigned one or two homework problems from
each section that we cover for an average of about 3 problems
per week. You may submit a solution more than once but for each
problem there will be a due date beyond which I will not accept
submissions.
You are allowed (and encouraged) to work together and
to use me as a resource while figuring out these problems but
there is to be no collaboration when writing up your
solutions. Also, always remember that failure to include
references is intellectual theft! Please see the
"Academic Honesty"
section of the Logger [3] to see how serious
this issue is to the university community.
You are to use the LATEX typesetting system to write up your
solutions. For those of you who are unfamiliar with LATEX, I
will be happy to give a brief workshop on how to install and use
this package. LATEX is now the expected way to typeset papers
in mathematics and science.
Since you will be using LATEX to submit your papers, it will be
easy to compile a collection of our solutions. So,
once you have a solution accepted (and if necessary, corrected),
you will submit the LATEX code to me in electronic form. I will
then collect the solutions into a single file that we can
"publish" to the class at the end of the semester. The immediate
benefit to you will be ongoing access to solutions of all homework
solutions (not just the ones you have worked) to use as learning
aids throughout the semester.
When working your problems, feel free to use (or not) any
technology that you like (e.g., SAGE, Scientific Notebook,
CABRI, Geometers Sketchpad, calculators, Mathematica,
MATLAB, etc.).
2.6 Reading
You need to read math to learn math.
2.7 Course Information Updates
If you
wish, I will post (and update) a grade report on your current
standing in the class on my university web page. To have your
information posted you need to print your name, the class
(MATH 420), and a code on a sheet of paper. Then sign the
paper and physically hand it to me. The code is to be a
sequence of up to 23 symbols I can type on a keyboard.
2.8 Total Points
Homework: | 64% |
Examinations: | 27% |
Final Examination: | 9% |
|
2.9 Ungraded Homework Assignment
Look over both my university web page
http://math.ups.edu/~bryans/ [1] and the course
webpage for MATH 420 you'll find there. Then send an e-mail message
to me at bryans [at] ups.edu that includes brief descriptions
of both your background in mathematics and any aspects of your
post-graduation
plans.
References
- [1]
-
Bryan Smith's Homepage
http://math.ups.edu/~bryans/
- [2]
-
Math 420A Course Webpage
http://math.ups.edu/~bryans/Current/Spring_2009/420Index_Spring2009.html
- [3]
-
University Academic Handbook (The Logger)
http://www.ups.edu/x4716.xml
- [4]
-
Location of MikTeX package for using LATEX
miktex.org
- [5]
-
Development tools for composing LATEX documents
miktex.org/Links.aspx
http://www.esm.psu.edu/mac-tex/
- [6]
-
"The Nuts and Bolts of Proofs", Antonella Cupillari
http://www.amazon.com/exec/obidos/tg/detail/-/0120885093/
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