Math 202A - Section 1 - Introduction to Topology and Analysis

Fall 2021

Instructor: Marc Rieffel

Lectures: TTh 9:40-11:00 pm, Valley Life Sciences 2040.

Course Control Number: 22271

Office: 811 Evans

Office Hours: Tuesdays 11-12, 1-2; Thursdays 11-12

GSI: Eric Jankowski

Office for office hours: 939 Evans

Office Hours: Mondays 10:30-12; Wednesdays 12:30-2

Prerequisites: Math 104 and considerable experience with other upper-division mathematics courses in dealing with quite abstract concepts and with constructing somewhat complicated proofs. Math 105, 110, 142 and 185 give especially useful preparation. I have no restrictions on enrollment by undergraduates. But undergraduates must fill out a form that can be found by going from the department home web page to "courses", then "enrollment" then "availability updates".

Recommended Texts (available free on-line):
Real and Functional Analysis 3rd ed. by Serge Lang, Springer-Verlag
Basic Real Analysis by Anthony Knapp, Birkhauser.
Advanced Real Analysis by Anthony Knapp, Birkhauser.
Analysis Now by Gert K. Pedersen, Springer-Verlag.
Measure Theory by Paul Halmos, Springer-Verlag (a classic).
Real Analysis for Graduate Students by Richard F. Bass.
Functional Analysis by Richard F. Bass.
General Topology by John L. Kelley (a classic).
General Topology, Chapters 1-4, by N. Bourbaki (the original classic)
Measure, Integration & Real Analysis by Sheldon Axler
Real Analysis by Bruce Blackadar. This is a preliminary version of a remarkable book-in-progress.

The Lang text gives a presentation of the material that is somewhat closer to that which I will give than do the other texts.

My understanding is that through an agreement between UC and the publishers, chapters of the texts by Lang, Knapp, Pedersen and Halmos are available for free download by students. You can find the chapters of the Lang text here, and chapters of the Knapp texts here, and here, chapters of the Pedersen text here, and chapters of the Halmos text here. You may need to use campus computers to authenticate yourself to gain access.
Links for the six free on-line books are:
Bass Real Analysis, Bass Functional Analysis, Kelley Topology, Bourbaki General Topology, Axler Measure Integration Real Analysis, and Blackadar Real Analysis. The link for ancient lecture notes of mine on measures and integration can be found at the bottom of my home web page.

Syllabus: This course, and Math 202B, are "tool courses", in that they cover some basic mathematical concepts that are of importance in virtually all areas of mathematics and its applications. (For lack of time we will not be able to present any of the applications, beyond a few hints.) In Math 202A we will cover: Metric spaces and general topological spaces, compactness, theorems of Tychonoff, Urysohn, Tietze, locally compact spaces; an introduction to general measure spaces and integration of functions on them, with Lebesgue measure on the real line as a key example; Banach spaces of functions, and the very beginnings of functional analysis. In Math 202B most of these topics will be developed further, especially measure and integration, and functional analysis.
In my lectures I will try to give careful presentations of the material, well-motivated with examples.

Grading: I plan to assign roughly-weekly problem sets. Collectively they will count for 50% of the course grade. Students are strongly encouraged to discuss the problem sets and the course content with each other, but each student should write up their own solutions, reflecting their own understanding, to turn in. Even more, if students collaborate in working out solutions, or get specific help from others, they should explicitly acknowledge this help in the written work they turn in. This is general scholarly best practice. There is no penalty for acknowledging such collaboration or help.
There will be a final examination, on Tuesday, December 14, from 3 to 6 pm, which will count for 35% of the course grade. There will be a midterm exam, at the regular class time, on Tuesday, October 19. It will count for 15% of the course grade. Here is a
sample midterm exam. There will be no early or make-up final examination. Nor will a make-up midterm exam be given; instead, if you tell me ahead of time that you must miss the midterm exam, then the final exam will count for 50% of your course grade. If you miss the midterm exam but do not tell me ahead of time, then you will need to bring me a persuasive doctor's note or equivalent in order to try to avoid a score of 0.

Comments: Students who need special accommodation for examinations should bring me the appropriate paperwork, and must tell me somewhat more than a week in advance of each exam what accommodation they need for that exam, so that I will have enough time to arrange it.

Problem sets: We will use GradeScope for posting the problem sets and for uploading the solutions.

The above procedures are subject to change.

Using TEX: I encourage students to write up their problem-set solutions in TEX, more specifically LATEX. (But I do not require this.) LATEX is a powerful mathematical typesetting program which is widely used in the sciences, engineering, etc., for documents that use a lot of mathematical symbolism. Thus learning to use TEX is a valuable skill if you work in such fields.

You can freely download versions of TEX onto your computer.
If you use Mac OS, you can find it at MacTex.
If you use Windows, you can find it at
proTeXt.

Several guides to using TEX are listed on the department's computer support web pages. Others can be found by searching on-line for "latex tutorial'' or "latex manual''.

The best way to start learning TEX is not by trying to compose the long header, but rather by having a file that already has a header, and then gradually modifying that file as you learn how TEX works.
To obtain such a TEX file with a header, click
LATEX-sample . Make a copy to play with, once you have downloaded TEX onto your computer. At first, don't modify anything above "\begin{document}". Since the output from TEX is a PDF file that can be directly uploaded to GradeScope, this would eliminate the need to scan pages of paper for uploading to GradeScope.

This page was last updated on 09/01/2021