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
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
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 ter
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.
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 metr
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 object
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, th
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 deco
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
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
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 f
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 sp
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
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 compac
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.