## Topology Reading Course, Fall Semester 2002, Georgetown U.

Dec. 6 Please look at the e-mail I sent you regarding the class.

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Dec. 6, 2002

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Reiss 252 office hours:

MTW 12:15-2:15
M  4:00-5:00
F 12:15-1:15

extended hours for last week of classes:

Dec. 2: 12:15 to 1, 4:30 to 6
Dec. 3: 12:15 to 6 (except about 2:15 to 3:15 when I'll be in a meeting)
Dec. 4: 12:15 to 2:15, 3:15 to 4:30
Dec. 6: 12:15 to 2:15
Dec. 7: afternoon

Office hours for Paul Kainen during study days and final exams

Dec. 7, 2 to 3pm; Reviews from 3:10 to 4:50; 5 to 7pm
Dec. 9, 2 to 6:30pm
Dec. 10, 2 to 7pm
Dec. 11, 3 to 6pm; Review for calculus 8:30 to 10:30pm
Dec. 13, 4 to 7pm
Dec. 14, 3 to 7pm
Dec. 16, 2 to 8pm
Dec. 17, 2 to 8pm
Dec. 18, 2 to 8pm

Papers and projects for combinatorics must be given to me
in person - so I can ask you a few questions about them.

Students in the topology seminar should send me e-mail
to set up a suitable time for your oral exam.

```

Here is a pointer to my page on topology .

Do problems 11 and 13 p. 95 and write them up. Give me your solution this week so I can return and discuss them next Monday. Also do the in-class exercise: f(C) subset of f(C') if C is a subset of C', and write it up as we did for similar problems in class.

Here are the three problems I listed today. They are simple but will be good exercises to help you get the basic set theory techniques that are used to prove things in topology.

```Suppose that f: X --> Y is a function. (1) Show that f^(-1)(A U B)
= f^(-1)(A) U f^(-1)(B), A,B subsets of Y. (2) Show that f^(-1)(f(C))
is a subset of C, where C is any subset of X. (3) How are the sets
f(f^(-1)(A)) and A related, where A is any subset of Y?
```

Write up problem #8 on p.95 and give it to me (or leave it in my mailbox in Reiss 256) sometime before next Monday - hopefully in the next few days - so I can give you some detailed critiques in your arguments. All the problem requires is a careful reading of what the various topologies are and what it means for a function to be continuous. So the write-up should be somewhat similar to my description of the solutions to #5 and #6 of the same section. Please try to be as clear and concise as possible in your answers.

Recall that we showed that f:X-->Y is continuous if and only if f^(-1)(tau_Y), which is the family {f^(-1)(U): U in tau_Y}, is contained in tau_X.

Also, read pp. 95 to 98, concentrating on Theorems A to F.

The course will cover Simmons' book on Topology, published by McGraw-Hill. The book is in the library in the math department lounge and I will give anyone copies of the necessary pages. Students are expected to become familiar with the basic concepts and methodology of point-set topology: separation properties, connectedness, and compactness, as well as subspaces, quotient spaces, and the properties of continuous mappings. Rather than aiming to cover a large amount of material, the emphasis will be on understanding these basics and being able to actually construct some of the proofs. Students will present material during class, in an interactive environment, and eventually I will test your understanding by giving an oral and/or written exam.

In the organizational meeting, to help everyone get started and since I didn't yet have the parts of the book xeroxed, I gave a brief lecture and defined what a topology is. See topology page.

(Namely, a family tau of subsets of a given nonempty set X which is nontrivial - containing both the empty set 0 and the whole set X - and is closed under finite intersections and arbitrary unions.) I also pointed out that a motivating example is take to X to be the real line (or (0,1) or any open interval), and tau to be the family of all subsets formed by taking arbitrary unions of open intervals. Since the intersection of the open intervals (-1/2^n,1/2^n) is {0}, one should not expect a topology to be closed under infinite intersections. Exercise 1. Find all possible topologies on the set {1,2,3}. For comparison, there is only one possible topology on {1}, while there are four different topologies on the set X = {1,2} (namely, {0,X}, {0,X,{1},{2}}, {0,X,{1}}, and {0,X,{2}}).

I also defined _relation_ and _equivalence relation_. A relation on a set X is a set of ordered pairs of elements from X. A relation R is reflexive if for all x in X, (x,x) is in R; R is symmetric if for all x,y in X, (x,y) is in R implies (y,x) is in R; R is transitive if for all x,y,z in X, (x,y) in R and (y,z) in R imply (x,z) in R. R is an equivalence relation if it is reflexive, symmetric and transitive. Examples. If X is the set of all integers, let R be the relation given by (x,y) in R if and only if y-x is divisible by 3. So, for instance, (7,13) is in R but (7,14) is not since 6 is divisible by 3 but 7 is not divisible by 3.

Exercise 2. Show that if X is the set of all differentiable functions defined on (0,1) with (f,g) in R provided that f' = g', then R is an equivalence relation on X. More generally, show that defining two functions to be related if their difference is a constant function is an equivalence relation.

In the second class, we counted the number of topologies on 3 points and found a total of 29. That is, excluding the trivial smallest topology (consisting of only the empty set and the entire set, called the ``co-discrete'' topology) and the trivial largest topology (which consists of _all_ subsets and is called the ``discrete'' topology), there are 27 nontrivial topologies. As one might expect, it is a multiple of 3.

I looked up on the web and found the following about the number of topologies on a set with n elements:

```1,4,29,355,6942,209527,9535241,642779354,63260289423
```
for n = 1,2,3,4,5,6,7,8,9, resp., according to the on-line encyclopedia of integer sequences (brought to us courtesy of Ma Bell. Note that while 355-2 is not divisible by 4, 6942-2 is divisible by 5, and 9535241-2 is divisible by 7, so maybe the number of nontrivial topologies on n points, when n is prime, is divisible by n.

Mostly for later use, I defined the induced function (let's call it P(f)) corresponding to a function f from a set S to set T. P(f) maps the power set P(S) to the power set P(T) by sending the subset E of S to the set {f(e): e in E} which is a subset of T. In general, P(f) does not map one topology to another, but we looked at a special situation where it does. Let S and T both be the set {1,2}, and let f be the function which interchanges 1 and 2. Then P(f) maps the discrete topology to itself, the codiscrete to itself, and interchanges the two nontrivial topologies.

In this sense, the set {1,2} has only three essentially distinct topologies. Or as we may say later, there are only three different ``homeomorphism types'' of topology on this set.

Exercise 3. How many distinct types are there for {1,2,3}?

I asked you to do the following for our next class on Sept. 16:

Exercise 4. (see p. 93, Example 4) Check that the set of all subsets of a set S which have a finite complement (together with the empty set if S is infinite) constitute a topology, called the cofinite topology. I mentioned in class that this is trivial if S is finite; in that case, the cofinite topology is the discrete topology. But I left out the condition that the empty set should be included in the general definition - a mistake, since otherwise the empty set might not be included as its complement is everything and that could be infinite. The exercise just requires you to check that the conditions of a topology are satisfied and uses some facts about finiteness, which I will let you discover ;-)

We did this in class and noted that we needed to use two identities for sets (the complement of the intersection is the union of the complements, and the complement of the union is the intersection of the complements), and also two facts about finiteness: the union of two finite sets is finite and any subset of a finite set is finite. If the set theory identities look strange, check them by showing that an element belongs to one side if and only if it belongs to the other.

For 9/23, on pp. 94-95, do problems 1,2,5 and 6 as Exercises 5 to 8. We did problems 1 and 2 in class but see if you can do them again on your own - without looking at the notes. For problem 5, we started the problem and noted that, for A subset of X and tau a topology on X, the family tau|A = {B| B = A intersect E, E in tau}, which is formed by intersecting A with all the elements of tau, is indeed a collection of subsets of A, and that tau|A must contain 0 = A I 0 (writing "I" for "intersection") and A = A I X.

For 9/30, do problems 7,8,9 as Ex. 9 to 11. Recall that a function f from X to Y is a continuous map of the topological space (X,tau) to (Y,sigma) if for any V in sigma, f^(-1)(V) is in tau. For any subset A of Y, f^(-1) is defined to be the set of all x in X with f(x) in A; that is, f^(-1)(A) is the _pre-image_ of A. So, using the standard terminology that an element of a topology is called an open set, f is continuous if the pre-image of each open set is also an open set.

As a warm-up, we proved that the inclusion function i from A to X, with i(a) = a, is continuous as a map from (A,tau|_A) to (X,tau): Let V be any element of tau. Then i^(-1)(V) = {a in A: i(a) in V} = A I V, which is an element of tau|_A.

It is ok for you to discuss the problems together and even to work out the answers jointly. But everyone should strive to understand each of them to the extent that you could put up the answers at the board.

Back to my classroom page . Further references: or topology page , and page for previous topology course. Dec. 6, 2002; pck