Math 535: Daily Log of Material Covered and Assigned Homework

Week	Date	Topics covered	References	Homework assigned	Homework due
1	8/27	Quick overview of topology and its applications. Topology of Rn. Metric spaces.	Willard § 1.2 Bredon § I.1 Munkres § 2.20 Brown § 2.8 Sections 1-5 of this entry.	HW1 - 0827 due 9/5
	8/29	Topological spaces. Continuous functions.	Willard § 2.3, 3.7 Bredon § I.2 Munkres § 2.12 Brown § 2.2, 2.5	HW1 - 0829 due 9/5
	8/31	Homeomorphisms. Neighborhoods. Bases and subbases. Comparing topologies.	Willard § 2.3, 2.4 Bredon § I.2 Munkres § 2.12, 2.13 Brown § 2.1, 2.6, 2.7	HW1 - 0831 due 9/5
2	9/3	Labor Day - No lecture.
	9/5	Subspaces. Products.	Willard § 3.6, 3.8 Bredon § I.3, I.8 Munkres § 2.16, 2.15 Brown § 2.4, 2.3	HW2 - 0905 due 9/12	HW 1: Lectures 8/27, 8/29, 8/31 Solutions
	9/7	Infinite products. Disjoint unions.	Willard § 3.8 Bredon § I.8 Munkres § 2.19 Brown § 3.1	HW2 - 0907 due 9/12
3	9/10	Quotient spaces.	Willard § 3.9 Bredon § I.13 Munkres § 2.22 Brown § 4.2	HW3 - 0910 due 9/19
	9/12	Limit points and closure. Interior. Sequences.	Willard § 2.3, 4.10 Bredon § I.3 Munkres § 2.17 Brown § 1.3, 2.9	HW3 - 0912 due 9/19	HW 2: Lectures 9/5, 9/7 Solutions
	9/14	More on sequences. Hausdorff spaces. Countability axioms.	Willard § 2.4, 4.10, 5.13, 5.16 Bredon § I.2, I.5 Munkres § 2.17, 4.30 Brown § 2.10	HW3 - 0914 due 9/19
4	9/17	Nets.	Willard § 4.11 Bredon § I.6 Munkres § 3.Supplement Sections 1-7 of this entry.	HW4 - 0917 due 9/26
	9/19	Compactness.	Willard § 6.17 Bredon § I.7 Munkres § 3.26, 3.27 Brown § 3.5	HW4 - 0919 due 9/26	HW 3: Lectures 9/10, 9/12, 9/14 Solutions
	9/21	More on compactness. Compactness in Rn. (Intro to) Compactness via nets.	Willard § 6.17 Bredon § I.7 Munkres § 3.26, 3.27 Brown § 3.5	HW4 - 0921 due 9/26
5	9/24	Compactness via nets. Zorn's lemma.	Willard § 6.17 Bredon § I.7 Munkres § 3.Supplement	HW5 - 0924 due 10/3
	9/26	Tychonoff's theorem.	Willard § 6.17 Bredon § I.8 Munkres § 5.37 Brown § 3.5	HW5 - 0926 due 10/3	HW 4: Lectures 9/17, 9/19, 9/21 Solutions
	9/28	Compactness and completeness in metric spaces. Uniform continuity.	Willard § 6.17 Bredon § I.9 Munkres § 3.27, 3.28 Brown § 3.6	HW5 - 0928 due 10/3
6	10/1	Separation axioms.	Willard § 5.13, 5.14, 5.15 Bredon § I.5 Munkres § 4.31, 4.32 Brown § 2.10	HW6 - 1001 due 10/10
	10/3	More on separation axioms. Urysohn's lemma.	Willard § 5.13, 5.14, 5.15 Bredon § I.5, I.10 Munkres § 4.31, 4.32, 4.33 Brown § 2.10	HW6 - 1003 due 10/10	HW 5: Lectures 9/24, 9/26, 9/28 Solutions
	10/5	Urysohn metrization theorem.	Willard § 7.22, 7.23 Bredon § I.9 Munkres § 4.34	HW6 - 1005 due 10/10
7	10/8	(Improved) Urysohn metrization theorem. Tietze extension theorem.	Willard § 5.15, 5.16, 7.23 Bredon § I.10 Munkres § 4.34, 4.35 Brown § 3.6	HW7 - 1008 due 10/17
	10/10	Locally compact spaces.	Willard § 6.18 Bredon § I.11 Munkres § 3.29 Brown § 3.6	HW7 - 1010 due 10/17	HW 6: Lectures 10/1, 10/3, 10/5 Solutions
	10/12	One-point compactification.	Willard § 6.19 Bredon § I.11 Munkres § 3.29 Brown § 3.6	HW7 - 1012 due 10/17
8	10/15	Stone-Čech compactification.	Willard § 6.19 Bredon § I.11 Munkres § 5.38 Brown § 3.6	HW8 - 1015 due 10/24
	10/17	Proper maps.	Bredon § I.7, I.11 Brown § 3.6	HW8 - 1017 due 10/24	HW 7: Lectures 10/8, 10/10, 10/12 Solutions
	10/19	Connectedness.	Willard § 8.26 Bredon § I.4 Munkres § 3.23, 3.24 Brown § 3.3	HW8 - 1019 due 10/24
9	10/22	Connected components. Path-connectedness. Path components.	Willard § 8.26, 8.27 Bredon § I.4 Munkres § 3.24, 3.25 Brown § 3.3, 3.4	HW9 - 1022 due 10/31
	10/24	Local path-connectedness. Homotopies.	Willard § 8.27, 8.32 Bredon § I.4, I.14 Munkres § 3.25, 9.51 Brown § 3.3, 3.4, 6.5	HW9 - 1024 due 10/31	HW 8: Lectures 10/15, 10/17, 10/19 Solutions
	10/26	Homotopy equivalence. Categories.	Willard § 8.32 Bredon § I.14 Munkres § 9.51, 9.58 Brown § 6.5, 6.1 Sections 1-4 of this entry.	HW9 - 1026 due 10/31
10	10/29	Functors. Pointed spaces.	Willard § 8.32 Bredon § I.14 Brown § 6.1 Sections 1-3 of this entry.	HW10 - 1029 due 11/7
	10/31	Relative homotopies. Intro to groupoids.	Willard § 8.32, 8.33 Bredon § I.14 Munkres § 9.51 Brown § 6.2	HW10 - 1031 due 11/7	HW 9: Lectures 10/22, 10/24, 10/26 Solutions
	11/2	Fundamental groupoid. Fundamental group.	Willard § 8.33 Bredon § III.2 Munkres § 9.52 Brown § 6.2	HW10 - 1102 due 11/7
11	11/5	Baire category theorem.	Willard § 7.25 Bredon § I.17 Munkres § 8.48	HW11 - 1105 due 11/26
	11/7	Applications of the Baire category theorem.	Willard § 7.25 Bredon § I.17 Munkres § 8.48, 8.49	HW11 - 1107 due 11/26	HW 10: Lectures 10/29, 10/31, 11/2 Solutions
	11/9	More applications of the Baire category theorem.	Willard § 7.25 Bredon § I.17 Munkres § 8.48, 8.49	HW11 - 1109 due 11/26
12	11/12	Partitions of unity and paracompactness. Note: Office hours canceled today and tomorrow.	Willard § 6.20 Bredon § I.12 Munkres § 6.41	HW12 - 1112 due 11/28
	11/14	More on partitions of unity and paracompactness.	Willard § 6.20 Bredon § I.12 Munkres § 6.41	HW12 - 1114 due 11/28	No HW due.
	11/16	Lecture canceled.
11/19 - 11/23		Thanksgiving break - No lectures.
13	11/26	Compact-open topology.	Willard § 10.43 Bredon § VII.2 Munkres § 7.46 Brown § 5.6, 5.9	HW13 - 1126 due 12/5	HW 11: Lectures 11/5, 11/7, 11/9 Solutions
	11/28	More on the compact-open topology. Evaluation map.	Willard § 10.43 Bredon § VII.2 Munkres § 7.46 Brown § 5.6, 5.9	HW13 - 1128 due 12/5	HW 12: Lectures 11/12, 11/14 Solutions
	11/30	Exponential law.	Willard § 10.43 Bredon § VII.2 Munkres § 7.46 Brown § 5.6, 5.9	HW13 - 1130 due 12/5
14	12/3	Convergence in function spaces.	Willard § 10.43 Bredon § VII.2 Munkres § 7.46	HW14 - 1203 due 12/12
	12/5	Compactly generated spaces.	Willard § 10.43 Munkres § 7.46 Brown § 5.9	HW14 - 1205 due 12/12	HW 13: Lectures 11/26, 11/28, 11/30 Solutions
	12/7	Continuity of limit functions.	Willard § 10.43 Munkres § 7.46	HW14 - 1207 due 12/12
15	12/10	Completeness of C(X,Y). Metrizability of C(X,Y).	Willard § 10.43 Munkres § 7.46, 7.47 This entry.	No HW assigned.
	12/12	Last lecture. Compactness in function spaces. Arzelà-Ascoli theorem.	Willard § 10.43 Munkres § 7.46, 7.47	No HW assigned.	HW 14: Lectures 12/3, 12/5, 12/7 Solutions
	12/13	Additional office hour 3-4 PM (Thursday 12/13).
	12/14	No lecture.
Monday 12/17		Additional office hour 4-6 PM.
Tuesday 12/18		Final exam, 1:30 - 4:30 PM, room 443 Altgeld (usual lecture room). Open book exam. You may bring: the four textbooks, notebooks, printouts of course material, and your own homework. Glossary. Solutions.

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