Math 467 (Honors Toplogy), fall 2019

Syllabus

General Topology References

COURSE LOG:

W 8/19 (F2F) From text: sectoons 12, 13, 15, 16, 17, 19, 30 (read)

Introduction of problems in problem set 1

PROBLEM SET 1

Lecture 8-19 (scan)

F 8/21(online) Problems in set 1 (start)

Problem 1.1 (Sam)

Problem 1.2 (Tariq)

Problem 1.1.5 (Kelechi)

Problem 1.3 (Jacob)

Problem 1.4 (Patrick)

3 topologies on the upper half plane (scan)

M 8/24 (online)

Problem 1.5 (Jun)

Problem 1.7.8 (Bryan)

Problem 1.9 (Bernardo)

Problem 1.15 (Matthew)

Also discussed: 14, 17, 18, 22, 23, 24.

Lecture 8-24 (scan)

(includes the example: "Hoelder spaces are not separable")

W 8/26 (F2F)

Discussion of: 16, 19, 20, 21, Hoelder spaces not separable.

Lecture 8-26 scan (5 pages)

F 8/28 (online)

Lecture 8-28 scan (3 pages)

Problem 1.10 (Eli)

Problem 1.11 (Tariq)

Problem 1.12 (Sam)

Problem 1.13 (Kelechi)

Extension to the closure, regular spaces (problem set 2)

PROBLEM SET 2

M 8/31 (online)--read Munkres sections 20, 21, 31, 32, 33

Problem 2.1 (Matthew)

Problem 2.7 (Jun)

Problem 2.8 (Bryan)

Problem 2.9 (Patrick)

Problem 2.10 (Jacob)

Problem 2.2 & remark on 2.8 (scan, 2 pages)

W 9/2 Problems 2.4 (Eli), 2.5 (F)

2.4 (Eli)

Normal spaces: problems 2.14, 2.11, 2.13 (F)

Urysohn's Lemma

Lecture 9-2 (scan, 4 pages)

F 9/4 (online)

2.6 (Larissa)

2.12 (Sam, F)

2.15 (Tariq)

2.16 (F)

Tietze extension (F)[Munkres, no. 35]

Lecture 9-4 (scan) 6 pages

M 9/7 (online) Baire property (#48 in [Munkres])

2.17 (Matthew)

2.18 (Kelechi)

2.19 (Jacob)

metric completion via embeddings

Notes on Baire spaces

Lecture 9-7 scan (3 pages)

PROBLEM SET 3

1-F 2-Bernardo 3-Eli 4-F 5, 5.5, 6, 7: review of compact spaces (open)

8-Patrick 9-Jun 10-Bryan 11-Jacob 12-Larissa 13-Tariq

14-Kelechi 15-Matthew 16-Sam 17-Eli 18, 19, 20-open.

W 9/9 (F2F) Hilbert cube, Urysohn metrization theorem(Munkres, no. 34)

Part I: embeddings into product spaces

Lecture 9-9 scan (4 pages)

F 9/11 (online) Urysohn metrization (Part II)/complete metrizability and G-delta sets

Lecture 9-11 scan (5 pages)

(includes 3 exercises)

M 9/14 (online) Problems from set 3 (students)

Lecture 9-14 (2 pages)

3.3 (Eli)

3.5, 3,6, 3.7 (Jacob)

W 9/16 (online) Compactness vs. sequential compactness

Lecture 9-16 (3 pages)

(Read in [Munkres]: no. 28, no. 45)

3.8 (Patrick)

3.9 (Jun)

3.10 (Bryan)

F 9/18 Problem set 4: Compactness and local compactness-(version: 10/5)

Lecture 9-18 (2 pages)

3.13 (Tariq)

3-14 (Kelechi)

M 9/21 Compactness (cont.)--Problems from set 3

3-15 (Matthew)

3.16 (Sam)

3-17 (Eli)

Lecture 9-21 (2 pages)

W 9/23 (in person)-Tychonoff's theorem

Lecture 9-23 (3 pages)

F 9/25 Spaces of maps of separable metric spaces with compact domain are separable/

Locally compact, connected metric spaces are separable. (Problems 12, 13 on compactness list.)

Lecture 9-25 (3 pages)

M 9/28 Locally compact spaces

Lecture 9-28 (3 pages)

from compactness-loc. compactness list (problem set 4, see 9/18):

6-Bernardo 9-Larissa 14-Sam 15--Kelechi 16-Eli 17-Jun 18-F 19-F 20-Tariq 21-F, 22-F

4-6 (Bernardo)

4-15 (Kelechi)

4-17 (Jun)

W 9/30 Compactifications in general, Stone-Cech (read [Munkres, no. 38])

Lecture 9-30 (4 pages)

Set 5: Notes and problems on compactifications (version: 10/5)

F 10/2 Complete metrizability (4 problems)

(repeats part of 9-11 lecture notes.) Goal: a subset of a complete metric space is

completely metrizable iff it is a G-delta subset.

Ex. 1-Jacob Ex. 2-Patrick Ex.3-Matthew Ex.4-Bryan

Ex.1-(Jacob)

Ex.2 (Patrick)

Ex. 3 (Matthew)

Ex. 4 (Bryan)

Heine-Borel metrics (paper: read it.)

M 10/5:proper maps (set 4, problems: 14, 19, 20, 21, 22)

Set 5 assignments:

1-F; 2+3: Eli 4: Bernardo 5: Sam 6: Kelechi 8: Jun 11: Matthew 12:Bryan

Lecture 10-5 (problems 4-19, 4-21, 4-22, 4-23)

4-20 (Tariq)

W 10/7 :Heine-Borel property/ locally compact normed linear spaces are finite-dimensional/

rigidity of compact+Hausdorff

Proper metric spaces (includes 3 problems)

Lecture 10-7 (4 pages)

F 10/9: problems from set 5

Lecture 10-9

5-4 (Bernardo)

5-5 (Sam)

5-8 (Jun)

(problems 5-1, 5-2, 5-3, 5-4, 5-5, 5-8)

M 10/12 problems: 5-6, 5-11, 5-12

from "proper metric spaces" handout: ex.1 (Patrick), ex.2 (Tariq), ex.3 (Jacob)

Lecture 10-12

5-11 (Matthew)

5-12 (Bryan)

Ex.1 (Patrick)

Ex.2 (Tariq)

Ex. 3 (Jacob)

W 10/14 MIDTERM (material up to 10/9)--online.

General Topology Review Map

(Make sure you know the definitions and equivalent conditions for each concept,

and review the proofs of the various implications.)

midterm

solutions

F 10/16 Spaces of maps, Ascoli's theorem [Munkres no. 46+47]

Ascoli-Arzela Notes (includes 14 problems=problem set 6)

Lecture 10-16

Assignments from Ascoli-Arzela notes:

Sam: Ex. 2, Ex. 4 Bernardo: Prob. 3, Prob.4 Eli: Prob. 7, Prob. 8

Kelechi: Prob 9 Patrick: Prob. 10 Jun: Prob. 11

M 10/19 Topologies in spaces of maps [Munkres 46+47]

Lecture 10-19

W 10/21 Stone-Weierstrass theorem (in person)

Problem Set 7

Patrick: 8/18. Matthew 7/7.5/7.7 Tariq: 2 Sam: 5/6 Bryan: 1/17

Eli: 9/10 Jacob: 12, 14 Kelechi: 19 Jun: 21/22 Bernardo: 20/23

Notes on Stone-Weierstrass (includes proof of Weierstrass approximation, and 3 problems)

F 10/23 Compact-open topology on spaces of maps

problems from Ascoli-Arzela notes

Ex. 2 (Sam)

Prob. 3, 4 (Bernardo)

Prob 8 (Kelechi)

Prob 10 (Patrick)

6-11 (Jun)

M 10/26 connectedness, path connectedness

[Munkres: 23-24-25]

Lecture 10/26

Problem set 8: connectedness

7-1, 7-17 (Bryan)

7-2 (Tariq)

7-5 (Sam)

7-6 (Sam)

7-7, 7-7.5, 7-7.7 (Matthew)

7-8 (Patrick)

7-12, 7-14 (Jacob)

7-18 (Patrick)

W 10/28 connected spaces, path connectedness

Ref: [Munkres, Willard]

Lecture 10/28

F 10/30 Cantor surjectivity theorem

Ref:[ Pugh, Willard]

Lecture 10/30

M 11/2 Problem session (from problem set 7)

Problems: 1, 2, 6, 7, 8, 12

W 11/4 Cantor spaces: Moore-Kline characterization theorem

Ref: [Pugh]

Lecture 11-4

Handuot 9: Cantor set, Cantor spaces (incomplete)

F 11/6 Peano spaces/more problems from set 7

Lecture 11-6

7-20, 7-23 (Bernardo)

7-9 (Eli)

7-10 (Eli)

7-19 (Kelechi)

7-21, 7-22 (Jun)

M 11/9 Quotient topology, quotient maps [Munkres no.22]/characterization results

Lecture 11-9

Topological Uniqueness of Spaces of Reals (paper)

Assignments from problem set 8:

2=Bernardo 6=Sam 9=Tariq 11=Kelechi 12=Matthew 13=Eli 14=Bryan 15=Jun 16=Patrick 17=Jacob

8-2 (Bernardo)

8-6 (Sam)

8-9 (Tariq)

8-11 (Kelechi)

8-12 (Matthew)

8-13 (Eli)

8-14 (Bryan)

8-16 (Patrick)

8-17 (Jacob)

W 11/11 Covering dimension/subsets of euclidean space/read: partition of unity [no.36]

Lecture 11-11

F 11/13 Covering dimension: embedding theorem

Lecture 11-13

M 11/16 Problem session (from problem set 8)

Connectivity of matrix groups

(alternative solution to problem 8-14)

W 11/18 Fundamental group [Munkres 52,53]

Lecture 11-18

F 11/20 Basic properties of fund. group and free homotopy

Lecture 11-20

M 11/23 Covering spaces and the fundamental group of the circle

Lecture 11-23

Practice problems from [Munkres]:

p. 335: 4, 5/p. 341: 1, 5, 6/p. 347: 7, 8

FINAL EXAM: Friday, December 4, 10:30AM--12:45PM (online)

Included in final: material since 10/16:

Topologies in spaces of maps: Arzela-Ascoli, Stone-Weierstrass

Connected and path-connected spaces

Cantor spaces: Cantor surjectivity thm, Moore-Kline Theorem, Peano spaces

Quotient topology, quotient maps, perfect maps

Covering dimension: subsets of euclidean space, embedding theoorem

Homotopic maps, fundamental group, covering spaces: first definitions and results

final exam solutions

Included in course (from [Munkres]):

Ch. 2: except 14

Ch. 3: all

Ch. 4: except 36

Ch. 5: all

Ch. 7: all

Ch. 8: except 49

Ch. 9: 51 to 54