3. Wed, Sept. 3
In calculus, we are also used to thinking of continuity in terms of convergence of sequences.
Recall that a sequence (xn ) in X converges to x if for every > 0 there exists N such that for all
n > N , we have xn 2 B (x). We say that a tail
23. Mon, Oct. 20
The next topic is one of the major ones in the course: compactness. As we will see, this is the
analogue of a closed and bounded subset in a general space. The denition relies on the idea of
coverings.
Denition 23.1. An open cover of X is
17. Mon, Oct. 6
(5) Similarly, we can think of Zn acting on Rn , and the quotient is Rn /Zn (S 1 )n = T n .
=
(6) The group Gl(n) acts on Rn (just multiply a matrix with a vector), but this is not terribly
interesting, as there are only two orbits: the or
5. Mon, Sept. 8
At the end of class on Friday, we introduced the notion of a topology, and I asked you to think
about how many possible topologies there are on a 3-element set. The answer is . . . 29. The next
few answers for the number of topologies on a
8. Mon, Sept. 15
Last time, we saw that if (an ) is a sequence in A X and an ! x, then x 2 A. But the converse
is not true in a general topological space. (The fact that these are equivalent in a metric space is
known as the sequence lemma.) To see this,
CLASS NOTES
MATH 551 (FALL 2014)
BERTRAND GUILLOU
1. Wed, Aug. 27
Topology is the study of shapes. (The Greek meaning of the word is the study of places.) What
kind of shapes? Many are familiar objects: a circle or triangle or square. From the point of vi
14. Monday, Sept. 29
On Friday, we introduced the idea of a coproduct, which is dual to the product. In the case of a
space X which happens to be the union of two open, disjoint, subspaces A and B, then the glueing
lemma told us that X satises the correct
20. Mon, Oct. 13
What we have done so far corresponds roughly to Chapters 2 & 3 of Lee. Now we turn to
Chapter 4.
The rst idea is connectedness. Essentially, we want to say that a space cannot be decomposed
into two disjoint pieces.
Denition 20.1. A disco
11. Monday, Sept. 22
Last time, we were talking about homeomorphisms.
Example 11.1.
(1) Consider tan : (0, ) ! (0, 1). This is a continuous bijection with
2
continuous inverse (given by arctangent)
(2) Consider ln : (0, 1) ! R. This is a continuous biject
39. Mon, Dec. 1
The last main topic from the introductory part of the course on metric spaces is the idea of a
function space. Given any two spaces A and Y , we will want to be able to dene a topology on
the set of continuous functions A ! Y in a sensible
42. Mon, Dec. 8
Last time, we were discussing CW complexes, and we considered two dierent CW structures on S n . We continue with more examples.
(2) RPn . Lets start with RP2 . Recall that one model for this space was as the quotient of D2 ,
where we impo
35. Monday, Nov. 17
Exam day.
36. Wednesday, Nov. 19
We nally arrive at one of the most important denitions of the course.
Denition 36.1. A (topological) n-manifold M is a Hausdor, second-countable space such that
each point has a neighborhood homeomorphi
32. Mon, Nov. 10
Last time, we saw that a space is normal if and only if any two closed sets can be separated by
a continuous function (modulo the T1 condition). Here is another important application of normal
spaces.
Theorem 32.1 (Tietze extension theore
29. Mon, Nov. 3
Locally compact Hausdor spaces are a very nice class of spaces (almost as good as compact
Hausdor). In fact, any such space is close to a compact Hausdor space.
Denition 29.1. A compactication of a noncompact space X is an embedding i : X
26. Mon, Oct. 27
Closely related to compactness is the following notion.
Denition 26.1. We say that a space X is sequentially compact if every sequence in X has a
convergent subsquence.
Example 26.2. The open interval (0, 1) is not sequentially compact be