MAT4500: Report from the lectures H-09

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:

See the message dated 18/8.

Thursday August 20th:

See the message dated 20/8.

Tuesday, August 25th:

I started in section 13 by proving Lemma 13.1 and 13.2. I also stated 13.3
(without proof) and used this Lemma to show that the topology in the
plane generated by open discs and by open rectangles actually is
the same topology. I also defined the lower limit topology and the K-topology
and compared these two topologies and the standard one.
Finally I completed section 13 by showing how a subbasis can be used to define a topology.
Then I started lecturing from $ 14 by defining the order topology on a set with a simple relation.

Thursday, August 27th:

I explained that the order topology on
the positive integers was equal the discrete topology.
Then I lectured $ 15 more or less as in the text book
(but I said nothing about Theorem 15.2).
I also followed the text book lecturing $ 16,
but dropped example 3 and Theorem 16.4 ( page 90/91).
Finally I started lecturing $ 17 and ended the lecture
by stating and proving Theorem 17.1.
I will proceed by lecturing $ 17 next Tuesday.

Tuesday, September 1 st :

I explained that an alternative definition of a topological space is to specify the closed sets as a family of subsets satisfying the conditions in Theorem 17.1, and I defined the Zariski topology as an example. I discussed closed sets in the subspace topology (Theorem 17.2, 17.3). Then I defined the notions of closure and interior of a subset and explained the Theorems 17.4 and 17.5. I introduced the notion of limit points,
gave some examples and explained Theorem 17.6. I then defined what it means that a sequence in a topological space is convergent, gave examples of spaces where the limit of such sequences was not necessarily unique and then I defined the Haussdorff property for a space.

Friday, September 4 th:

i gave some examples of topological Haussdorff spaces and some examples of spaces
without this property. I proved Theorem 17.8, defined the T1 property and proved Theorem 17.9. I explained that an infinite topological space with finite complement topology (which I sometimes call the cofinite topology) is a T1, but not a Haussdorff space (and I claimed that the Zariski topology also has this property). I proved theorem 17.10. Then I started to lecture $ 18, defined what we mean by a continuous map between two spaces, and I proved Theorem 18.1. I defined and characterized homeomorphisms. open and closed maps and gave some examples. Next Tuesday, I will finish $ 18, complete $ 19 and hopefully start to lecture $ 20.

Tuesday, September 8 th:

I gave another example of a homeomorphism namely stereographic projection from a circle minus one point to the real line, and explained that this construction generalized to all dimensions. I then proved that compositions of continuous maps were continuous. I proved the pasting lemma (Theorem 18.3), but I did not make any other comments about the remaining constructions of continuous functions (p. 107-11).
I then moved into $ 18, defined the box and product topology on infinite products and explained the difference of these topologies. I explained or more or less proved the theorems 19.4-6. I moved into $ 20 and recalled the definition of the metric topology.
I stated and proved Lemma 20.2 and explained why the euclidean metric and the square metric gave the same topology on R^n. Friday I will proceed with $ 20, and (hopefully) lecture $ 21 and parts of $ 22.

Firiday, September 11 th:

I proceeded lecturing from $ 20, defining a bounded metric space. I defined the uniform metric and uniform topology on R^J (J an index set) and compared this topology with the box and product topologies (Theorem 20.4) . I also explained that the product topology on R^\omega was induced by a metric (Theorem 20.5). I started lecturing from $ 21, stating and proving 21.1-3.

Tuesday, September 15 th:

I completed $ 21 by stating and proving Theorem 21.6. I then started lecturing from $ 22, motivating the definition of the quotient topology, by giving a couple of standard examples of spaces defined by certain equivalence relations on some given topological
spaces. I then defined and analyzed quotient maps and defined the quotient topology on a set of subsets defining a partition of a topological space (and thus the set of equivalence classes of an equivalence relation on a space) . I then stated Theorem 22.1., and stated and proved 22.2 and 22.3. Then started lecturing from $ 23. defining
a connected space and subspace of a space, and finally I stated and proved Lemma 23.1

Friday, September 18 th:

I completed $ 23 following the textbook, but dropping the examples on p. 151.
Then i lectured $ 24 also following the textbook, but dropping the examples 4, 5 and 6
(but I explaining example 7). I will start lecturing $ 25 and hopefully parts of $ 26 on Tuesday.

Tuesday, September 22 nd:

I started on $ 25, defining the connected components of a space, explaining their properties (Theorem 25.1). I also defined path components and explained Theorem 25.2 . Then I explained the concepts local connectedness and local path connectedness. I gave various examples of spaces that were connected or path connected but not locally connected or path connected. Then I proved the theorems 25.3-5. I then started on $ 26, defining a compact space and a compact subset of a space. I gave some examples. I proved then proved Theorem 26.2., gave an examples of a (non-Hausdorff) space where there were compact but not closed subset as a contrast to Theorem 26.3 which i formulated. I will start proving this theorem on Friday.

Friday, September 25 th:

A started proving Theorem 26.3 that a compact subset of a Hausdorff space is closed.
I then lectured the remaining part of section 26 more or less following the textbook
(but I explained a the Lemmas 26.4 and 26.8 rather scarcely merely pointing out how they followed from the proof of 26.3 and 26.7.
I then started on section 27. stating Theorem 27.1 and proving this theorem when X was the real numbers. I then ended the lecture by proving Theorem 27.3. I will proceed lecturing from $ 27 next Tuesday.

Tuesday, September 29 th.

I started by proving Theorem 27.4, and I lectured the rest of $ 27 following the textbook. I however skipped the proof of Theorem 27.7. I then lectured the whole of $ 28 (apart from Example 2 and 3)

Friday, October 2 nd.

I lectured $29. Apart from example 3, which I dropped, I followed the exposition in the textbook.

Tuesday, October 6 th

I lectured from $ 30 and followed the textbook, but I dropped example 3 and 4. The I started lecturing from $ 31 following the textbook. I ended the lecture by stating and proving Lemma 31.1. I will complete $ 31 and also (hopefully) lecture the whole of
$ 32 on Friday.

Friday, October 9 th

I explained Example 1 page 197, stated Theorem 31.2 and proved part (b) of this Theorem. Then I explained that a similar Theorem is not true for normal spaces and that example 2 and 3 gives a counterexample showing that products of normal spaces
are not necessarily normal. I lectured $ 32 stating and proving the theorems 32.1-3.
As an example I explained that R^\omega is normal (since this space is metrizable), but stated that R^J is not normal when J is uncountable. I did not say much about Theorem 32.4.

Tuesday, October 13 th

I lectured $ 33 following the textbook only skipping Example 1, p. 212

Friday, October 16 th.

I proved The Urysohn Metrization Theorem (Theorem 34.1). I gave only the first version of the proof (where the space is imbeddded in R^\omega with product topology).
Then I lectured the rest of $ 34. I stated Tietze Extension Theorem (without proof). (This is not part of the syllabus but I think you should know important Theorem).
I also start to talk about theThychnoff theorem. Again this is not part of the syllabus but you should also know this theorem. I will probably proceed with this the first part of the lecture next Tuesday and then I will start on $ 45.

Tuesday, October 20th.

I explained the proof of the Tychnoff Theorem (Theorem 37.2) (this is not part of the ordinary syllabus,
but the theorem is important and you should be aware of its content). I then moved to $ 43 and followed the textbook and ended by proving Theorem 43.5.
I will proceed with $ 43 on Friday and the start lecturing $ 45 (skipping $ 44).

Friday, October 23 rd.

I completed $ 43 by proving the Theorems 43.6 and 43.7. I then started on $ 45 explaining of how Theorem 45.1 followed from Theorem 28.2 and results from MAT 1300, Analysis 1 . I then explained the notion of equicontinuity and proved the Lemmas 45.2 and 45.3.

Tuesday, October 27 th.

I explained the notion of pointwise boundedness (for a family) and proved Theorem 45.4 and Corollary 45.5. I started on $ 46 explaining that the product topology can be viewed as a topology of pointwise convergence of functions. I then defined an explain the topology of compact convergence and how it related to the uniform topology and the topology of of pointwise convergence. I will proceed with $ 46 on Friday.

Friday, October 30 th.

I proceeded lecturing from $ 46, I did not say anything about compactly generated topological spaces (so I skipped 46.3-46.6) I stated Theorem 46.7 and stated and proved 46.8-46.11 following the text-book. I then turned to $ 51 defining the notion of homotopi and proved Lemma 51.1. I will proceed with $ 51 on Tuesday.

Tuesday, November 3rd.

I explained that any two maps into a convex subset of R^n are always homotopic. Then I considered two paths such that the end point of the first was equal the initial point of the second, and I defined the product of these path. I proceeded proving the important Theorem 51.2 and also Theorem 51.3. I then started on $ 52 giving a very short remainder of fundamental and elementary facts from group theory. I then defined the fundamental group of a topological space relative to a base points, and I showed that the groups relative to two different base points in the same path components are isomorphic (proving Theorem 52.1 and Corollary 52.2). I then gave the definition of a simply connected space.

Friday, November 6 th.

I proved that when a space is simply connected any two paths with the same start and endpoint are homotopic (Lemma 52.3). I explained how maps between spaces induces maps between the corresponding fundamental groups and proved Theorem 52.4. and Corollary 52.5. I explained the notion of covering maps and spaces explained Theorem 53.1, I proved Theorem 53.2 and 53.4 and also formulated (as an introduction to the next lecture) the path lifting Lemma (54.1).

Tuesday, November 10 th.

I started $ 54, by proving the path lifting and path homotopy lifting lemmas (Lemma 54.1 and 54.2). I proved Theorem 54.3, and explained why this gives a map from the fundamental group of the base space (of a covering map) into the fiber over the base (in the total space). I proved Theorem 54.4 giving some of the properties of this map.
I then proved that the fundamental group of the circle is isomorphic to the additive group of integers. Finally I stated Theorem 54.6 and proved part (a) of this statement. I will finish the proof of this theorem on Friday and then start lecturing from $ 55.

Friday, November 13 th.

I finished the proof of Theorem 54.6 and turned to section 55. I followed the textbook
stating and proving the Lemmas and Theorems given in 55.1-6.
I will finish $ 55 proving Corollary 55.7 on Tuesday. I will skip $ 56-57, and then start lecturing (and probably finish) $58.

Tuesday November 17 th.

I proved Corollary 55.7. Then I started lecturing from $ 58, proving Theorem 58.2, defining deformation retracts and the stating and explaining Theorem 58.3. I gave examples of deformation retracts (Example 1-3 p. 362). I defined the notion of homotopy equivalence and stated and proved 58.4-58.7.

Friday November 20 th.

I proved that the spheres of dimension >1 were simply connected (Theorem 59.1)
I then turned to $ 60 and followed the text book stating and proving the various statements about fundamental groups of some compact surfaces
(60.1-60.6). On Tuesday and Friday I will look at some old examproblems
(2007-2006).

Published Aug. 25, 2009 3:30 PM - Last modified Nov. 23, 2009 4:58 PM