First Day Handout:
Professor: Thomas Schlumprecht
Office: Blocker 625 Tel: 845-8840
Hours: We 9:00 - 10:00 am, Th 3:00 - 4:00 pm
E-mail: schlump@math.tamu.edu
Homepage: http://www.math.tamu.edu/~thomas.schlumprecht
- General Description
Construction of the real and complex numbers; topology of metric spaces, convergence, continuity,
compactness and conectedness; Cauchy sequences, completeness, Theorem of Baire;
introduction to point set topology. (For more detail see below)
- Textbook (for Math446 and Math447)
The required textbooks are
General Topology Schaum, McGraw-Hill 0-703-7988-2
Principles of Math Analysis Rudin, McGraw-Hill 0-070-5423-X
Grading
Course grades will be determined by one midterm, one final
and homeworks (33.3% each).
Copyright Statement: All handouts
and web-documents associated with this course are protected
by US Copyright law. One personal copy (or download) of any
of these documents is allowed by each student. Multiple copies
or sale of any of these materials is strictly forbidden.
Chapters
- Review of the real number line R,
- complete ordered field
- sequences, convergence, continuity
- open, closed, and compact sets in R
- Cauchy sequences, completeness
- Theorem of Heine Borel, Theorem of Bolzano Weierstrass
- Construction of R
- Short overview of some basics of set theory
- notations
- relations, functions, injectivity, surjectivity
- cardinalities of sets, uncountability of R, Schroeder-Bernstein theorem
- Metric Spaces
- Definition
- Examples
- convergence, continuity in metric spaces
- open, closed, and compact sets in metric spaces
- Cauchy sequences, completeness in metric spaces
- Fixed Points, Banach's fixpoint theorem and applications
- The Baire Cathegory Theorem and applications
- Topological Spaces
- Definition of generel topological spaces, open and closed sets, convergence
of sequences
- Continuity in
- .....
Midterm: October 19 th
Homeworks (Due on Thursdays)