On this page I shall give a short report from the lectures. The notes are primarily intended for students who have not been able to attend the class, but would like to see what has been covered.
Tuesday August 19th:
I covered the material in the "Brief review of sets and function" (more or less) through the section called "Functions". On Thursday I shall talk about countability from the same notes and then continue with chapter 2 of the textbook.
Thursday August 21st:
I first finished the "Brief review of sets and functions" by going through the material on countability, and then turned to section 12 in the book. Here I defined topologies and showed some examples (in addition to those in the book, I proved that the open sets in a metric space form a topology). I then turned to section 13 where I introduced the notion of a basis for a topology and proved that all bases generate topologies. On Friday I shall start with Lemma 13.1.
Friday, August 22nd:
I first completed section 13 by proving Lemma 13.1 and showing how a subbasis can be used to define a topology. I then defined neighborhoods and convergence of sequences, and tried to use these concepts to give an intuitive idea of how topologies describe "how close together" points are. I then rather quickly covered $$14, 15 and 16. Tuesday I shall talk on the important $17.
Tuesday, August 26th:
I first introduced closed sets (after some motivation from metric spaces) and proved Theorem 17.1. I then introduced the closure of a set and proved Theorem 17.5. Then we turned to limit points and proved Theorem 17.6 and its corollary. Next I defined Hausdorff spaces and proved Theorems 17.8 and 17.10. I ended the lecture by trying to motivate the abstract definition of continuous functions in $18.
Thursday, August 28th:
Today I covered most of $18 on continuous functions. I first defined continuity and then proved Theorem 18.1 (which I broke down into three separate propositions). I said a a few words about homeomorphisms and imbeddings, and then pointed out that to prove that a function is continuous, it suffices to show that the inverse image of all sets in a basis or a subbasis are open. I proved Theorem 18.4 as an illustration of this technique. Next time I shall probably prove the pasting lemma (Theorem 18.3) before I move on to $19.
Tuesday, September 2nd:
I first proved the pasting lemma from section 18 and then turned to section 19. Here I introduced the box topology and the product topology, and tried to show through an example why the product topology is more useful and less pathological than the box topology. I then proved Theorem 19.5 and Theorem 19.6. Toward the end I turned to $$20-21 on metric topology. I recalled the definition of the topology generated by a metric and defined metrizable spaces. I pointed out that different metrics may generate the same topology (e.g. different norms on R^n generate the same topology). I also pointed out that not all topological spaces are metrizable (e.g. a metrizable space is necessarily Hausdorff). Since most of $$20-21 is already known, I'll start with $22 on Friday — this means we are slightly ahead of schedule.
Friday, September 5th:
Today I covered most of $22 on quotient topologies. I first tried to motivate the theory by introducing the idea of a topology induced by a mapping or by an equivalence relation. Next I introduced quotient maps, and proved that they can be characterized as the continuous, surjective functions which map open, saturated sets to saturated sets. I then introduced quotient topologies and proved that they really are topologies. I then turned to examples and spent some time on the construction of a quotient topology on the torus (example 5 in the book). Finally, I proved the first part of the first part of Theorem 22.1. Next time I shall prove Theorem 22.2 and 22.3, and then start $23.
Tuesday, September 9th:
I finished the material on quotient topologies by first proving that the composition of two quotient maps is a quotient map and that a bijection is a quotient map if and only if it is a homeomorphism. I then used these results to prove Theorem 22.2 and its corollary (only the first part). I then started Chapter 3 by talking informally about connectedness and path connectedness before I defined connected spaces and proved Lemma 23.1, Lemma 23.2 and Theorem 23.3.
Friday, September 12th:
I continued $23 by proving theorems 23.4, 23.4 and 23.5. I then turned to $24 where I proved all the results. Finally, I talked a little bit about Example 7, where I proved that the set is connected, but omly indicated how one can prove that it is not path connected. On Tuesday I begin $25.
Tuesday, Sepiember 16th
Today I went through $25 following the book rather closely. On Friday I shall begin $26 on compactness.
Friday, September 19th:
I went through §26 on compactness with the exception of Theorem 26.9 (which I didn't have time for, but which I shall return to next time). In addition to what is in the book, I sketched the proof that a closed, bounded interval in R is compact (this is basically the proof of Theorem 27.1, which I shall not return to).
Tuesday, September 23rd:
I started by saying a few words about Tychonov's Theorem which says that the product of an arbitrary family of compact sets is compact (in the product topology). This theorem is not on the curriculum (but in the book, see §37), but important to know about.
I then introduced the finite intersection property and wrote down Theorem 26.9, leaving the proof as an exercise. I went rather quickly through §27 as most of it is well-known from MAT1300, just picking up the extreme value theorem (27.4) and the Lebesgue number lemma (27.5). I then introduced limit point compactness and sequential compactness, and proved Theorems 28.1 and Theorem 28.2.
Sigurd Segtnan will lecture on Friday and he will start with §29 on local compactness.
Friday, September 26th
Sigurd Segtnan lectured on §29 (local compactness). As an example he showed that the one-point-compactification of a compact space is not connected. Tuesday, October 2nd, we shall start chapter 4.
Tuesday, September 30th
A rather lazy lecture today which only covered §30 on countability axioms. I put in the proof of Theorem 30.1, but did not dig into the examples on Lindel?f spaces. We continue with §31 on Friday.
Friday, October 3rd:
I started by defining regular and normal spaces, and proved Lemma 31.1. Then I stated Theorem 31.2 and proved part b), before turning to Theorem 32.1 and its proof. I shall continue next time with theorems 32.2 and 32.3, and will then probably turn to section 33.
Tuesday, October 7th:
I proved theorems 32.2 and 32.3 and Urysohns's lemma (Theorem 33.1). Next time I shall finish §33 and then prove Urysohn's metrization theorem.
Friday, October 10th:
I defined completely regular spaces and wrote down Theorem 33.2 (without proof) before I turned to Urysohn's Metrization Theorem (Theorem 34.1). I described the theorem and gave an outline of its proof before turning back to §20 and the proof of theorem 20.5 (metrizability of R^N). I then reurned to §34 where I first proved "Step 1" as a separate lemma, and then finished the proof of the Urysohn Metrization Theorem (the "first version" in the book).
Tuesday, October 14th
I first stated (without proof) Tietze's extension theorem (§35). This theorem is not part of the syllabus, but it is definitely useful to know of its existence! I then went to §43 and §45 to pick up a few definitions and results that we shall need in the sequel: In §43 I introduced the uniform metric (page 266), proved Theorem 43.5 and (sketched the proof of) Theorem 43.6. I then pointed out that if X is compact, we can use the sup-norm (page 268) on C(X,Y). In §45 I pointed out that Theorem 45.1 is a consequence of results in MAT 1300 and Theorem 28.2.
After the break I turned to §45 where I proved lemmas 45.2 and 45.3. Next time (remember that the lecture is moved from Friday to Thursday) I'll start with Ascoli's theorem (Theorem 45.4).
Thursday, October 16th:
I first proved Ascoli's theorem (version 45.4), and then started §46 where I introduced the topology of pointwise convergence and sketched the proof of Theorem 46.1. I then turned to the topology of uniform convergence on compacts and theorem 46.2.
Tuesday, October 21st:
I dropped the bit on compactly generated spaces (lemma 46.3-theorem 46.7) and went straight to the compact-open topology where I proved Theorem 4.6.8, Theorem 4.6.9 and Theorem 4.6.11). Next time I start chapter 9.
Friday, October 24th:
I covered §51 emphasizing the ideas rather than the reparametrization details. Next lecture is on Tuesday October 30th (Tuesday is problem session) and will cover §52 and perhaps a little of §53.
Thursday, October 30st:
I spent most of the time reviewing group theory (a slightly extended version of what I said is available here ). I then introduced the fundamental group and proved Theorem 52.1.
Friday, October 31st:
Finished §52 and started §53. In the latter I put the emphasis of an intuitive understanding of covering spaces, and I spent most of the time on the covering spaces of the circle (in the form of a helix) and the torus (as in Example 4). I finished with the formal definitions of "evenly covered" and "covering space".
Tuesday, November 4th:
I finished §53 and proved lemma 54.1 (which I would call a theorem!)
Friday, November 7th:
Covered the material from 54.2 up to and including 54.5. I will not cover the material in the rest of this section (unless it turns out that we shall need it later).
Tuesday, November 11th:
Covered §55 up to and including Theorem 55.5. Remember that the next lecture is on Thursday.
Thursday, November 13th:
I first talked about theorems 55.6 and 55.7 and then turned to §58 where I got as far as homotopy equivalences (the definition on top of page 363).
Tuesday, November 18th
I defined homotopy equivalences and proved Lemma 58.4,Corollaries 58.8 and 58.6, and Theorem 58.7. I then turned to §59 where I proved Theorem 59.1 and Corollary 59.2.
Friday, November 21st:
I started by proving that the n-sphere is simply connected for n>1 and then turned to §60 where I proved all the results (cheating a bit in the proof of Theorem 60.3 by not checking in detail that p is really a covering map). This means that we have completed the syllabus and will use the last two lectures for review.
Tuesday, November 25th:
I started the review by recalling the basic notions of a topology, open and closed sets, and continuous functions. I then mentioned how we can build topologies from bases and subbases, and recalled subspace topologies, product topologies, quotient topologies and metric topologies, and said a few words about how we in practice prove that sets are open or closed and functions are continuous (here I mentioned pointwise continuity and the pasting lemma). I then went on to talk on connedtedness: its definition, the definition of components, that products and closures of connected sets are continuous, and that unions of connected sets are connected when they have at least one point in common. I then turned to compactness where I mentioned the definition, the alternative definition in terms of closed sets ("families with the finite intersection property") and the fact that products of compact sets are compact. Finally, I briefly mentioned the countability and separation axioms, Urysohn's lemma and Urysohn's metrization theorem.
Friday, November 28th:
I continued the review by briefly mentioning the completness of C(X;Y) (Theorem 43.6) and Ascoli's theorem before turning to Chapter 9 and algebraic topology. I defined homotopies and path homotopies and introduced the fundamental group, mentioning that in path connected spaces, this group is independent of the base point. I then introduced covering spaces, and recalled the ways in which paths and homotopies can be uniquely lifted to the covering space (including the fact that when the covering space is simply connected, there is a bijection between the fundamental group and the fiber of the base point, Theorem 54.4). I then described how we can "compare" fundamental groups using retracts, deformation retracts, homotopy equivalences and theorems 59.1 and 60.1. As we had some extra time I covered Example 2 on page 362 as an example of deformation retracts,