This page has material for the 1st semester and, further below, for the 2nd semester course.
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:
Hand-in by sliding your exam u n d e r 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.
In a previous incarnation of this course (the old syllabus (pdf) is available), the students found the following documents useful.
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 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.
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.
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) .
Here is a problem: Can you construct a complete metric on the
irrationals, equivalent to the usual metric?
Indeed, one can characterize
completable subsets of a metric space (pdf);
but don't look at this until you've seriously tried the above problem.
One tool used in topological dynamics is the Ellis enveloping semigroup (pdf). 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 (pdf)
by mimicking Cantor diagonalization for a general product. Perhaps you
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 (pdf).
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.
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.)
Any compact manifold can be embedded in a finite-dimensional Euclidean space (pdf).
Below, are some of the solutions submitted by former students.
Victor Brennan has determined the cardinality (ps) of the set of continuous functions from the reals to itself; along the space way he recapitulates Troy's cardinality argument.
Vic also proves that the sup-norm is complete (ps). and gives a nice summary of convergence facts.
He proves that a separable metric space is countably generated (ps). The version here needs a little polishing —its idea is correct.
Vic shows that a complete metric space is a BaireCatSpace (ps).
Kevin Dezfulian shows that the exponentiation realsnaturals is bijective with the reals. (Unfortunately, his solution was on his account, which disappeared.)
Jason Riedy constructed a complete metric on R minus a finite set of points. He seems close to constructing a metric on the set, I, of irrationals, equivalent to the usual metric, which makes I complete.
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.
These are all take-home (open book/notes) exams. They are to be slid u n d e r 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):
Jeremy Smith is keeping track of the new H-problems (html).
NOTE: Munkres uses a hyphen in naming a section of his text,
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
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