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Aut1997: MTG4302 3279 Elements.Topology.1 MWF6 LIT201

Aut1997: MTG5316 3280 Intro.Topology.1 MWF6 LIT201

This page has material for the
*1 ^{st} semester* and, further below, for the

Our text is Topology (2^{nd} edition).

Author: | James Munkres | ISBN: | 978-0131816299 |

Year: | 2000 | Publisher: | Prentice Hall |

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:

- Elementary Descriptive Set Theory.
- Introduction to Knot and Link Theory.
- Introduction to manifolds and algebraic topology.
- Introduction to Topological Dynamics and Ellis's Enveloping Semigroup.
- The Borel hierarchy. The Baire hierarchy.
- Elementary continua theory.

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.

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.

- A takehome midterm (pdf) (consisting of extensions of the problems in the notes).
- A list of problems (pdf) that -while much much longer than the actual exam- includes most of the problems that appeared on the final exam (pdf).
- An answer sheet (pdf) for the exam, although this does not give solutions.

*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) .

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.

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.

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 reals^{naturals}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.

Spr1998: MTG4303 5045 Elem.Topology.2 MWF6 LIT203

Spr1998: MTG5317 6171 Intro.Topology.2 MWF6 LIT203

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,
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.